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Differential geometry on finitely presented groups and related combinatorial invariants I. K. Babenko University of Montpellier, France
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    1. IntroductionMany classical concepts in differential geometry, such as Riemannian metric, geodesic, or volume, defined originally on manifolds, are easily transferred to simplicial polyhedra. In this connection a natural question arises: to what extent do global invariants, such as systolic volume, volume entropy, and some others, depend only on the fundamental group of the polyhedron in question? When methods of differential geometry are used on two-dimensional polyhedra with a given fundamental group, this formulation of the question allows one to obtain new invariants of finitely presented groups. This idea is not new and was apparently first stated by Gromov in [30], where the systolic area of a finitely presented group was actually introduced. Over the last 20 years significant progress has been made in the study of the systolic area of finitely presented groups. At the same time, this has led to a number of interesting problems that remain unsolved to this day. One aim of this survey is to introduce these problems to the general reader. The concept and some properties of the systolic area of a group are discussed in § 3. A completely different type of invariant of differential-geometric nature, naturally related to groups, was also originally discovered on closed manifolds. This is so-called volume entropy, also known as ‘asymptotic volume’. The idea originally appeared in Manning [47], who estimated from below the topological entropy of geodesic flow on a closed Riemannian manifold in terms of the exponential growth rate of the volume of a ball of large radius on the universal covering of this manifold. This growth rate is called the volume entropy of the Riemannian metric in question (see § 10 for the precise definitions). The volume entropy depends on the growth of the fundamental group of the manifold, and in the case of subexponential growth it is equal to zero. If the fundamental group of a manifold has exponential growth, then the topology of the whole manifold becomes significant, and a natural question arises: under what topological conditions on the manifold the lower bound of the volume entropies over all unit volume metrics is positive? This leads to the concept of the minimum volume entropy of the manifold in question. In fact, the smooth structure plays a secondary role in the definition of volume entropy. As in the case of systolic area, the only thing required for the definition of volume entropy is the ability to calculate lengths and volumes, and these concepts are well defined for $\operatorname{PL}$-metrics on an arbitrary polyhedron. Therefore, we arrive at the concept of the minimum volume entropy of a finitely presented group by considering all possible Riemannian $2$-polyhedra with fixed fundamental group and minimising the resulting volume entropies over the entire family of such polyhedra (see § 11). In the same section we consider some properties of the volume entropy of groups, as well as a number of examples. The volume entropy of groups (in all of its variants, see below) is a fairly new invariant, so there are still many more questions than answers. We have tried to include many of these questions in the text. Although this approach to the definition of the volume entropy of a group seems quite natural, an alternative view of this concept was presented in [17]. The main motivation for the alternative version of the minimum volume entropy is hyperbolic manifolds. The problem of the positivity of the minimum volume entropy for hyperbolic manifolds was known for a long time, and was successfully solved in [38] for surfaces and in [12] in the general case. The alternative definition of the volume entropy of a group applies only to geometrically finite groups. This definition consists of considering all finite aspherical polyhedra with fixed fundamental group of dimension equal to the geometric dimension of the group under consideration. The further steps are always the same: the minimum volume entropies of the polyhedra under consideration are minimised over the set of all such polyhedra. We discuss briefly this type of volume entropy in § 14. By a well-known theorem proved by Stallings (and, independently, Swan), groups of geometric (respectively cohomological) dimension 1 are precisely free groups. In this case both definitions of minimum entropy coincide trivially. The difference between the definitions of volume entropy becomes clear if the geometric dimension of the group is strictly greater than two (see [51], where a specific example of this difference is analysed). For groups of geometric, and therefore also cohomological dimension two the question of whether the two definitions coincide is an open problem (see § 14). It turns out that the differential-geometric invariants of groups discussed above are closely related to new combinatorial invariants that appeared recently and are of independent interest. It was noted in [3] that for large values of the systolic area of a group it is well approximated by a purely combinatorial invariant, called in [3] the ‘simplicial complexity’ of this group. Perhaps a more appropriate name for this invariant would be the ‘combinatorial area’ of the group. This is the term we use in this paper. The concept of the combinatorial area of a group has a transparent geometric meaning: it is the smallest number of 2-simplices contained in a 2-complex with the fixed fundamental group (the precise definitions and basic properties are presented in § 4). Therefore, a group is free if and only if its combinatorial area is zero. There are a number of open questions relating to combinatorial area, many of which are parallel to the corresponding problems for systolic area. This parallelism is not accidental, and we will try to follow it as much as possible in the presentation that follows. The connections of combinatorial area with other combinatorial invariants of groups are discussed in § 5. Section 6 explains how combinatorial area can be used to approximately find systolic area. Although these estimates are approximate, they give a good idea of the asymptotic behaviour of systolic area for large values of the parameter, for example, for cyclic groups in dependence on their order. The combinatorial area of a group is calculated from a minimal triangulation of some 2-polyhedron with fixed fundamental group. As recently proved in [16], for surface groups everything reduces to minimal triangulations of the corresponding surfaces. For certain genera, special complex structures emerge on these surfaces. The role and place of these structures are not clear; these issues are discussed in § 7. Another natural combinatorial invariant of a finitely presented group is the smallest number of vertices in a 2-complex fixed fundamental group. This invariant is of purely topological nature, as explained in § 8. Interest in this old topological idea was recently attracted by Karoubi and Weibel [37], so in [4] we called this invariant the Karoubi–Weibel complexity (briefly $\operatorname{kw}$-complexity) of the group. Karoubi–Weibel complexity is a new, as yet little-studied, invariant which also raises a number of open questions; this invariant is discussed in § 8. In § 9 we consider the connections of $\operatorname{kw}$-complexity with systolic area and volume entropy. As always, free groups stand apart. This is logical, since a free group is a purely one-dimensional object. Thus, systolic area in this case should be replaced by ‘systolic length’, and asymptotic volume by ‘asymptotic length’. For reasons of unification, in the latter case we still preserve the term ‘volume entropy’. In the case of a free group all differential geometry reduces to combinatorial questions on weighted graphs, a popular and well-studied topic in graph theory. Section 15 contains an analysis of the above invariants in the case of free groups. In conclusion, we note that for free groups many things become trivial and almost all questions discussed in our paper have a solution. The only exception is systolic length; its asymptotic behaviour in its dependence on the rank of the group is still not quite clear (see § 15). 2. Riemannian polyhedraConsider a finite polyhedron (a finite simplicial complex) $X$, which is always assumed to be connected. A Riemannian metric (or $\operatorname{PL}$-metric) $g$ on $X$ is a set of $C^{\infty}$-metrics defined on each simplex of the polyhedron under consideration. Here we assume that each metric $g_{\Delta}$ is a positive definite quadratic form with $C^{\infty}$-coefficients in the barycentric coordinates of $\Delta$. To avoid unnecessary discussions about the regularity of the form $g_{\Delta}$ on the boundary of $\Delta$, we assume that $g_{\Delta}$ is defined in some neighbourhood of the simplex $\Delta$. In addition, we assume, although this is not necessary, that the following consistency condition holds for the family (1):
$$
\begin{equation*}
\text{if}\ \ \Delta_1 \quad\text{and}\ \Delta_2 \text{ are two simplices of} \ X \ \text{and} \ \Delta_1\subset \Delta_2 ,\ \ \text{then} \ \ g_{\Delta_1}=g_{\Delta_2}\big|_{\Delta_1}.
\end{equation*}
\notag
$$
The metric $g$ on $X$ defines the distance $\varrho_g$ by
$$
\begin{equation}
\varrho_g(p,q)=\inf_{\gamma(t)}\int_0^1\|\gamma'(t)\|_g\,dt,
\end{equation}
\tag{2}
$$
where $\gamma(t)$ ranges over the family of piecewise linear curves $\gamma\colon [0,1] \to X$ connecting the points $p=\gamma(0)$ and $q=\gamma(1)$. It is well known that $X$ with the distance (2) is a geodesic space. Globally, geodesic spaces $(X,\varrho_g)$ can have a rather intricate structure, but if part of a geodesic passes through the interior of some simplex, then it is a $C^{\infty}$-curve, as in classical differential geometry. Note that in this paper we are not particularly interested in the structure of geodesics themselves. If $(X,g)$ is a Riemannian polyhedron of dimension $m$, then its volume, denoted by $\operatorname{Vol}(X,g)$, is equal to the sum of the volumes of all $m$-simplices calculated in the metric $g$. This volume coincides with the $m$-dimensional Hausdorff measure of the metric space $(X,\varrho_g)$. We use the notation $\operatorname{Vol}(X,g)$ in all dimensions, including $m=1$ and $2$, where it means the length and area, respectively. The systole of a polyhedron $(X,g)$ is defined by
$$
\begin{equation}
\operatorname{sys}(X,g)=\inf_{\gamma(t)}\int_0^1\|\gamma'(t)\|_g\,dt,
\end{equation}
\tag{3}
$$
where the infimum is taken over the family of piecewise linear closed ($\gamma(0)=\gamma(1)$) non-contractible curves $\gamma\colon [0,1] \to X$. Therefore, the systole is defined only for non-simply connected Riemannian polyhedra; it is easy to see that it is positive for any Riemannian metric $g$ on $X$. Now we can define the systolic volume of an $m$-dimensional polyhedron $X$ by
$$
\begin{equation}
\sigma(X):=\inf_{g} \frac{\operatorname{Vol}(X,g)}{(\operatorname{sys}(X,g))^m}\,,
\end{equation}
\tag{4}
$$
where $g$ ranges over the whole family of Riemannian metrics on $X$. The quantity (4) does not depend on the metric and is a topological invariant of $X$. For generic polyhedra $X$ the dependence of $\sigma(X)$ on the topology of $X$ is quite complicated and has not yet been fully elucidated. Nevertheless, the fundamental question of when $\sigma(X)$ is positive has already been answered. The following definition goes back to Gromov [29]. Consider the characteristic map defined uniquely up to homotopy.Definition 2.1. A polyhedron $X$ of dimension $m$ is called essential if its characteristic map $f_X$ is not homotopic to a map whose image lies in the $(m-1)$-skeleton of the space $K(\pi_1(X),1)$. Otherwise, $X$ is called inessential. The notion of essentiality originally appeared in the context of manifolds (see [29]). In this case essentiality can be expressed in purely homological terms, using the fundamental class. In the case of general polyhedra there is no fundamental class, but the mechanism of essentiality remains the same: the canonical map cannot be deformed to a skeleton of lower dimension. Example 2.2. $1^{\circ}$. A two-dimensional polyhedron $X$ is essential if and only if $\pi_1(X)$ is not a free group (see [30]). $2^{\circ}$. Let $X$ be a closed $m$-dimensional orientable manifold. Then it is essential if and only if
$$
\begin{equation*}
(f_X)_{\ast}([X]) \ne 0\quad\text{in}\ H_m\bigl(K(\pi_1(X),1),\mathbb{Z}\bigr).
\end{equation*}
\notag
$$
Here $[X]$ is the fundamental class of the manifold. $3^{\circ}$. Let $X$ be a closed $m$-dimensional non-orientable manifold. If where $[X]_2$ is the fundamental class modulo $2$, then $X$ is essential.It is precisely the homological conditions in examples $2^{\circ}$ and $3^{\circ}$ that were taken as the definition of the essentiality of a manifold in the original paper [29]. In this case the sufficiency of these conditions in the sense of Definition 2.1 becomes obvious. The necessity in $2^{\circ}$ was proved in [2]. The necessity in $3^{\circ}$ was also proved there, where it was formulated in terms of fundamental classes with local coefficients (see [2] for details). Calculating the systolic volume of an essential polyhedron is an extremely difficult problem even in fairly simple examples. It is known precisely only for three polyhedra, and all of them are surfaces. Example 2.3. $1^{\circ}$. A 2-torus: $\sigma(T^2)=\sqrt{3}/2$ (Loewner, 1949, unpublished, but well known). The extremal metrics on $T^2$ are the flat metrics corresponding to the hexagonal lattice. It is worth noting that this was the first result that gave rise to all systolic geometry (see [11]). $2^{\circ}$. The projective plane: $\sigma(\mathbb{R} P^2)=2/\pi$ (Pu [53], 1952). Extremal metrics in this case are metrics of constant curvature. $3^{\circ}$. A Klein bottle: $\sigma(K^2)=2\sqrt{2}/\pi$ (Bavard [10], 1986). This last example is remarkable in that the extremal metrics in this case are not smooth, contrary to naive expectations. Up to proportionality, the extremal metric is obtained by gluing two Möbius bands of curvature $+1$, symmetric with respect to the equator, where the edge of the band has constant latitude $45^{\circ}$. Thus, this example is a Riemannian polyhedron, although homeomorphic to a manifold. The gluing line of the Möbius bands is an embedded circle, along which the metric is not smooth, but Lipschitz. The systolic geometry of surfaces is the best studied branch of systolic geometry. Without discussing the achievements of this direction here, we refer the reader to [40], where they can find many beautiful results and elegant examples. 3. Systolic area of a finitely presented groupAll groups considered in this paper are assumed to be finitely presented, unless otherwise stated. With this in mind, we will simply say ‘group’ in what follows. In [30] Gromov established the following remarkable fact. Theorem 3.1. There exists a positive constant ${\mathbf c} $ such that for any polyhedron $X$ of dimension $2$ with non-free fundamental group the inequality holds. The value of the optimal constant in this inequality is unknown. Gromov’s technical estimate ${\mathbf c} \geqslant 10^{-4}$ has been improved many times since then. See Conjecture 3.6 below and the comment there. Theorem 3.1 justifies the following definition. Definition 3.2. The systolic area of a group $G$ is the quantity where the infimum is taken over all $2$-polyhedra $X$ with fundamental group $G$. Although the definition of the systolic area of a group has been known for 30 years, the following question remains fully open even in the case of the simplest non-free group $G=\mathbb{Z}_2$ (see also Conjecture 3.6 below). Conjecture 3.3. For any non-free group $G$ there exists a systolic minimal polyhedron, that is, a $2$-polyhedron $P$ such that Note that there is no hope for the uniqueness of such a polyhedron, even if it exists. Definition 3.2 and Theorem 3.1 immediately imply that the equality $\sigma(G)=0$ holds if and only if the group $G$ is free. Moreover, for any non-free group the inequality $\sigma(G) \geqslant {\mathbf c}$ holds. Remark 3.4. To bring a non-trivial meaning to Definition 3.2 for free groups, it should be slightly modified, which we do in § 15. The following proposition reflects some properties of systolic area; below the symbol $\ast$ denotes the free product of groups, and $\mathbb{F}_n$ is a free group of rank $n$. Proposition 3.5. $1^{\circ}$. Let $G_i$, $i=1,2$, be two groups. Then In particular, for any group $G$ and any natural integer $n$ $2^{\circ}$. Let $G$ be a group and $H < G$ be a subgroup of index $k$, then The simplest non-free group is $\mathbb{Z}_2$. It is well known (see, for example, [40]) that among all non-simply connected surfaces the projective plane has the smallest systolic area. This leads us to the following long-known conjecture (see [56]). Conjecture 3.6. $1^{\circ}$. For any $2$-polyhedron $X$ with non-free fundamental group, the inequality holds. $2^{\circ}$. Moreover, if $\pi_1(X) \ne \mathbb{Z}_2 \ast \mathbb{F}_n$, then this inequality is strict. It seems quite plausible that both parts of this conjecture are true. Its proof would give the precise value of the optimal constant in Theorem 3.1: ${\mathbf c}=2/\pi$, and automatically a lower bound for the values of $\sigma(G)$ over all non-free groups. Despite the fact that Conjecture 3.6 has been known for a long time and has not yet been verified, there are a number of results on lower bounds for the universal constant ${\mathbf c}$ (see [41] and [56]). As far as the author knows, the strongest estimate to date, was obtained in [39].Although Proposition 3.5 is quite elementary, it leads to another, quite non-trivial conjecture, also formulated in [56]: This conjecture clarifies the appearance of the factor $\mathbb{F}_n$ in the second part of Conjecture 3.6. Conjecture 3.7 is non-trivial, as its solution requires more than an analysis of specific polyhedra. Namely, let $G$ be a group, and let $X$ be some $\varepsilon$-optimal polyhedron, that is,
$$
\begin{equation*}
\pi_1(X)=G\quad\text{and}\quad \sigma(X) < \sigma(G)+\varepsilon.
\end{equation*}
\notag
$$
Then, of course, the polyhedron $Y=X\vee S^1$ has the required fundamental group $\pi_1(Y)=G \ast \mathbb{Z}$ and satisfies the inequality $\sigma(Y) < \sigma(G)+\varepsilon$. However, there is no reason to claim that $Y$ is $f(\varepsilon)$-optimal for the group $G \ast \mathbb{Z}$, where $f(\varepsilon)$ is some function tending to zero with $\varepsilon$. Whether (7) is true or not, Proposition 3.5 shows that the set of isomorphism classes of groups with systolic area at most $T$ is always infinite if $T \geqslant 2/\pi$. The trivial reason for this is the possibility of adding free factors of the form $\mathbb{F}_n$. This leads to the following definition. Definition 3.8. A group $G$ has free index $0$ if it cannot be represented in the form $G=H\ast \mathbb{Z}$. Note that the subgroup $H$ in this decomposition is generally not uniquely defined (see [42]). The problem of finiteness for the number of groups of bounded systolic area was posed by Gromov in the most general form in [30]. As we see, without reasonable restrictions on the properties of groups there can be no hope for the finite set of groups of bounded systolic area. In this connection we denote by $\mathcal{G}_{\sigma}(T)$ the set of isomorphism classes of groups of $G$ that have zero free index and such that $\sigma(G) \leqslant T$. As usual, if $\mathcal{X}$ is a set, then $|\mathcal{X}|$ denotes its cardinality. The finiteness problem was successfully solved by Rudyak and Sabourau [56]. They proved that the set $\mathcal{G}_{\sigma}(T)$ is always finite, and the following estimate for its cardinality holds: where $A$ is an explicit constant. This upper bound was significantly improved in [3], where it was proved that for all $T\geqslant 2$ the inequality
$$
\begin{equation}
|\mathcal{G}_{\sigma}(T)| \leqslant B^{T^{1+B'/\sqrt{\ln(B'T)}}}
\end{equation}
\tag{9}
$$
holds, where $B$, $B'$, and $B'$ are explicit constants. It is worth noting that all constants arising in this area of geometry, except for the optimal ones, in spite of being explicit, are cumbersome and therefore ‘unattractive’. The reader can find one example of such a constant in (15), where it looks more or less acceptable. In order not to obscure the essence of the issue, these technical constants are designated by letters. The reader interested in details will always find the explicit form of these constants in the references provided. When studying the set $\mathcal{G}_{\sigma}(T)$ as a function of $T$, questions of two types arise that have not yet been fully answered. The first type concerns groups that actually occur in the set $\mathcal{G}_{\sigma}(T)$ for a given $T$. Answering this question is not easy even for small $T$. For example, as we saw in (6), the set $\mathcal{G}_{\sigma}(1/4-10^{-10})$ is empty. However, $\mathcal{G}_{\sigma}(4)$ is already quite large, and contains completely unexpected groups. Here effective upper bounds for $\sigma(G)$ are obtained by producing concrete Riemannian polyhedra with fundamental group $G$. Omitting the computational details, we can say that the set $\mathcal{G}_{\sigma}(4)$ contains a number of surface groups (for example, the fundamental groups of non-orientable surfaces of genus $\leqslant 16$ and orientable surfaces of genus $\leqslant 8$), as well as the groups
$$
\begin{equation*}
\mathbb{Z}_n, \quad n\leqslant 5; \qquad \mathop{\ast}\limits_{i=1}^n(\mathbb{Z}_2)_i, \quad n\leqslant 6; \qquad \mathbb{Z}_3 \ast \mathbb{Z}_3; \quad\text{and}\quad \mathbb{Z}\oplus \mathbb{Z} \oplus \mathbb{Z},
\end{equation*}
\notag
$$
among others. The group $\operatorname{PSL}(2,\mathbb{Z})$ is also contained in $\mathcal{G}_{\sigma}(4)$, but it is not known whether $\operatorname{SL}(2,\mathbb{Z})$ belongs to it. Another type of questions related to the set $\mathcal{G}_{\sigma}(T)$ arises in the study of groups with very large systolic area, that is, $\sigma(G) \gg 1$. A new combinatorial invariant of groups was defined in [3], which we call here the combinatorial area of the group. It turns out that if $\sigma(G) \gg 1$, then $\sigma(G)$ is fairly well approximated by the combinatorial area of $G$, which, in turn, allows us to obtain estimate (9). We proceed to the definition of the combinatorial area and the description of some of its properties in the next section. 4. Combinatorial area of a finitely presented groupFor a finite polyhedron $X$ we denote the number of its $k$-dimensional simplices by $s_k(X)$. Definition 4.1. Let $G$ be a finitely presented group. Then the combinatorial area ($\kappa$-area) of $G$ is the quantity where the infimum is taken over the $2$-polyhedra $X$ with fundamental group $G$. A polyhedron $P$ satisfying the conditions is called a minimal polyhedron of the group $G$.To avoid various unnecessary complications, we assume that no minimal polyhedron contains vertices incident to a unique $1$-simplex. It is also reasonable to assume that a minimal polyhedron contains at most two consecutive isolated edges. In general, such a polyhedron is not unique (even if it contains no isolated edges at all), but the number of minimal polyhedra is always finite. Remark 4.2. The quantity (10) was originally introduced in [3] and called there the simplicial complexity of a group. We think that the ‘combinatorial area’ of a group is a more appropriate name, and we stick to it in this paper. For obvious reasons, computing the $\kappa$-area of a group is simpler than computing its systolic area. However, as the value of $\kappa(G)$ increases, the complexity of calculations, even computer-assisted ones, grows exponentially. First of all, we note that, as for systolic area, $\kappa(G)=0$ if and only if $G$ is a free group. The smallest positive value taken by $\kappa$-area on non-free groups is $10=\kappa(\mathbb{Z}_2)$, and the group $\mathbb{Z}_2$ is the only group of $\kappa$-area 10 among all groups with zero free index. In this sense, a full analogue of Conjecture 3.6 holds for $\kappa$-area. All known exact calculations of the invariant $\kappa$ can be reduced to the following two examples. We introduce the following notation. Let $M_h^{\pm}$ be a surface of genus $h$, where the sign $+$ or $-$ is chosen depending on whether or not the surface is orientable. Let Example 4.3. The initial values of the $\kappa$-area correspond to the following groups (see [3]): The question mark in the table means that there can be other groups of $\kappa$-area $17$ apart from $\mathbb{Z}_3$. The author does not know whether or not such groups actually exist, but the inequality $17 \leqslant \kappa(\mathbb{Z}_2 \ast \mathbb{Z}_2) \leqslant 18$ makes us cautious. In addition to this table, we note that the minimal polyhedron for the group $\mathbb{Z}_2$ is unique and is obtained by taking the quotient of an icosahedron by central symmetry; it coincides with the minimal triangulation of $\mathbb{R} P^2$. In the case $G=\mathbb{Z}\oplus \mathbb{Z}$ the minimal polyhedron is also unique and coincides with the minimal triangulation of the torus (see [3]). These minimal triangulations are shown in Fig. 1. Here centrally symmetric points on the boundary of the circle are identified to obtain $\mathbb{R} P^2$, and the fundamental domain of the torus is glued using translations by the vectors corresponding to endpoints of the blue path $\alpha$ and green path $\beta$. It is easy to see that these vectors generate a hexagonal sublattice of the original lattice in which the triangles are taken. Remark 4.4. The calculations of Example 4.3 were carried out in [21] by painstaking manual case-by-case enumeration. This explains why it is not possible to make much progress in the calculation of the values of $\kappa$-area using this method. It would be interesting to use computer calculations to describe groups that have a small $\kappa$-area. The concept of a ‘small’ $\kappa$-area is very conditional here and depends both on the hardware capabilities and the possible algorithmic difficulties. Example 4.5. For surface groups the $\kappa$-area was completely calculated in the elegant paper [16]. In particular, Borghini and Minian showed there that for non-simply connected surfaces ($h\ne 0$) the equality holds, where $\delta(M_h^{\pm})$ is the number of $2$-simplices in the minimal triangulation of the surface $M_h^{\pm}$. In fact, it was proved in [16] that every minimal polyhedron of the group $\pi(h)^{\pm}$ is homeomorphic to the surface $M_h^{\pm}$. Therefore, to obtain the final numerical formulae for the $\kappa$-area of surface groups, it remains to apply the now classical formulae of Ringel [54] and Jungermann–Ringel [35], expressing $\delta(M_h^{\pm})$ in terms of the genus $h$ of the surface. We denote the least natural number greater than or equal to $x$ by $\lceil x\rceil$. Then we have and Formula (11) shows, in particular, that an orientable surface of genus 2 is singular not only from the point of view of algebraic geometry, but also in the combinatorial sense. In addition to the anomalously large number of simplices needed to triangulate this surface, there exists a minimal triangulation of combinatorial diameter $3$ on it, although on all other orientable surfaces a minimal triangulation has the combinatorial diameter at most two. (By the combinatorial diameter of a polyhedron we mean the maximum distance between its vertices in the edge metric on the $1$-skeleton.) We return to triangulations of a sphere with two handles in § 9. The behaviour of combinatorial area is in many ways similar to that of systolic area. In particular, the analogue of Proposition 3.5 is valid in a slightly stronger form. Proposition 4.6. $1^{\circ}$. Let $G_i$, $i=1,2$, be two groups. Then here $u=0$ if at least one of the groups $G_i$ is free, and $u=1$ otherwise. In particular, for any group $G$ and any natural integer $n$ $2^{\circ}$. Let $G$ be a group and $H < G$ be a subgroup of index $k$. Then As already mentioned, an analogue of Conjecture 3.6 holds for $\kappa$-area. This follows, in particular, from the calculations in [21]. The following conjecture is analogous to Conjecture 3.7 and also remains completely open. This conjecture looks no less plausible than Conjecture 3.7; furthermore, it is valid for surface groups. This follows from the result of Borghini and Minian [16] stated below. Theorem 4.8. For any surface group $G=\pi(h)^{\pm}$ and an arbitrary group $T$ the inequality holds. In particular, $\kappa(G\ast \mathbb{Z})=\kappa(G)$. 5. Relationship of combinatorial area with other combinatorial invariants of groupsThere are two other combinatorial invariants of groups that are related to combinatorial area. These are the so-called Delzant $T$-invariant (see [24]) and the $c$-complexity of Matveev and Pervova (see [48] and [52]). Both are defined in a similar way. Consider a group $G$ and some presentation
$$
\begin{equation}
\mathcal{P}(G)=\langle a_1,a_2,\dots,a_n \mid r_1,r_2,\dots,r_m\rangle
\end{equation}
\tag{14}
$$
of $G$. Let $|r_i|$ denote the length of the word $r_i$ in the alphabet $\{a_i\}_{i=1}^m$, and let be the length of the presentation. Definition 5.1. The $c$-complexity of a group $G$ is defined as the length of the most economical presentation: Similarly, the $T$-invariant of a group $G$ is defined by The definition of $T(G)$ can look odd at first glance, but it is quite natural. Each group can be defined only by relations of length $2$ and $3$, a so-called triangular presentations. The invariant $T(G)$ is just the minimum number of relations of length $3$ in a triangular presentation. It is obvious from the definition that both invariants, the $c$-complexity and the $T$-invariant, are semi-additive with respect to the free product of groups. The same applies to $\kappa$-area. A surprising property of the $T$-invariant is that it is purely additive. This means that for any groups $G_1$ and $G_2$ the following equality holds (see [24]): At the same time, the $T$-invariant is completely insensitive to $2$-torsion, for example, $T(\mathbb{Z}_2 \ast \mathbb{Z}_2 \ast \dots \ast \mathbb{Z}_2)=0$, and $T\bigl(\operatorname{PSL}(2,\mathbb{Z})\bigr)=T(\mathbb{Z}_3)=1$. It is obvious from the definition that for any group $G$ the inequality holds. A further comparison of the three invariants, namely, $c$-complexity, $T$-invariant, and $\kappa$-area, is made in the following proposition (see [3] and [52]).Proposition 5.2. For any group $G$ the following inequalities hold: where $h$ is the minimum number of relations needed to define $G$. If $G$ does not have $2$-torsion, then If $G$ does not have free factors isomorphic to $\mathbb{Z}_2$, then 6. Using combinatorial area to calculate systolic areaWe have already mentioned above that for large values of systolic area, $\sigma(G) \gg 1$, it is approximated fairly well by $\kappa$-area. The following comparison result was established in [3]. In a slightly weaker but simpler form inequality (15) looks like this: for any $\varepsilon > 0$, if $\sigma(G)$ is sufficiently large, then
$$
\begin{equation*}
2\pi\sigma(G) \leqslant \kappa(G) \leqslant (\sigma(G))^{1+\varepsilon}.
\end{equation*}
\notag
$$
Estimate (15) plays a key role in the proof of (9). Although it looks asymptotically quite satisfactory, as the degrees of the left- and right-hand sides asymptotically coincide, a natural question arises: is there a more subtle asymptotic connection between the functions $\sigma(G)$ and $\kappa(G)$ when $\sigma(G)\gg 1$? To formalise this question, we give the following definition. Definition 6.2. Let $f_i(t)$, $i=1,2$, be two positive functions of positive argument. We say that they are asymptotically equivalent, written $f_1\sim f_2$, if there exist two constants $0 < c \leqslant C$ such that for some $t_0$. Remark 6.3. This definition differs significantly from the definition of the equivalence of growth functions (see, for example, [31]), although there is some similarity between them. For example, $2^t \nsim 3^t$ from the point of view of Definition 6.2. We also note that all functions arising in applications are bounded away from zero and bounded on any compact interval of the positive half-line, so the constants $c$ and $C$ can be chosen for the whole common domain of the functions $f_i(t)$, $i=1,2$. Example 6.4. $1^{\circ}$. Formulae (11) and (12) imply that $2^{\circ}$. For the systolic area of a surface $M_h^{\pm}$, as a function of genus $h$, the equivalence holds.$3^{\circ}$. It turns out that a similar equivalence holds for the systolic area of surface groups: Despite its apparent simplicity, the equivalence (16) is a very serious result in the systolic geometry of surfaces, obtained by the efforts of different authors over a fairly long period of time. The lower asymptotic estimate in (16) was found by Gromov [29], and the upper one was obtained by Buser and Sarnak [22] from quite different considerations. For more information, see [40]. The equivalence (17) is also not at all obvious; it follows from the following result of Balacheff, Parlier, and Sabourau [7], which is of independent interest: Theorem 6.5. For a non-trivial group $G$ of free index zero, where $C$ is a universal positive constant, and $b_1$ denotes the first Betti number. Combining parts $1^{\circ}$ and $3^{\circ}$ of Example 6.4, we obtain the following equivalence for surface groups:
$$
\begin{equation*}
\sigma(\pi(h)^{\pm}) \sim \frac{\kappa(\pi(h)^{\pm})}{(\ln \kappa(\pi(h)^{\pm}))^2}\,.
\end{equation*}
\notag
$$
This leads us to the following conjecture. To keep in line with the estimates in (15), we can apply to (18) the function $t(\ln t)^2$, which is asymptotically inverse to $t/(\ln t)^2$. This gives us an equivalent formulation of Conjecture 6.6: As we have already noted, the exact values of the $\kappa$-area are known only in the cases listed above. Nevertheless, it is often possible to obtain fairly satisfactory estimates of this invariant. The following result was proved in [3]: Theorem 6.7. For any finite Abelian group $G$ the following double inequality holds: where $s$ is the number of invariant factors of $G$. This implies the following assertion: The upper bound in Corollary 6.8 is obtained using the Möbius telescope construction (see [3] for details). For the special group orders $m=2^n+1$, this construction yields a better upper bound than the one above: The table in Example 4.3 shows that for the group $\mathbb{Z}_3$ ($m=3$) the telescope is a minimal complex. So it is natural to ask the following question.Some other examples of the computation of $\kappa$-area can be found in the paper [3] cited already. In it the reader can also get acquainted with the technique of minimal triangulations, which was developed and widely used in that paper. For example, using this technique, it is easy to calculate that if $G_A$ is a right-angled Artin group, then
$$
\begin{equation*}
b_2(G_A) \leqslant \kappa(G_A) \leqslant 14 b_2(G_A),
\end{equation*}
\notag
$$
where $b_2$ denotes the second Betti number of the group. Good estimates for the $\kappa$-area of specific groups lead to fairly nice estimates of the systolic area. In this way, equipping the Mobius telescope with a special metric (see [3]) we obtain the inequality
$$
\begin{equation*}
\sigma(\mathbb{Z}_m) \leqslant \frac{1+2\sqrt{3}}{\pi}(1+\log_2m)
\end{equation*}
\notag
$$
for all $m \geqslant 2$. A similar metric, but with curvature $(\pi/2)^2$, of an arbitrary $2$-polyhedron, is described in detail in the proof of Theorem 13.1 below. A lower bound for the systolic area of a cyclic group can be obtained using Theorem 6.1 (see [3] for details). Let $\varepsilon > 0$; then for any sufficiently large integer $m$ the inequality
$$
\begin{equation*}
(\log_2m)^{1-\varepsilon} \leqslant \sigma(\mathbb{Z}_m)
\end{equation*}
\notag
$$
holds. Bounds for the systolic area of an arbitrary finite Abelian group are also obtained in [3]. To avoid cluttering the text, we do not present them here.
7. Minimal triangulations and complex structuresExample 4.5 shows that the combinatorial area of a surface group is completely determined by a minimal triangulation of the surface. We restrict ourselves to the orientable case. There is a complex structure naturally associated with each triangulation $\theta$ of an orientable surface $M_h$ of genus $h$. The standard way to obtain a complex structure is to define a Euclidean metric on each $2$-simplex of the triangulation such that the simplex is an equilateral triangle with edge length $1$. All such metrics on $M_h$ glue together to form a global flat metric with conical singularities. Singularities occur only at vertices of the triangulation, and the curvature at a vertex is equal to $2\pi-(\pi/3)\nu$, where $\nu$ is the valency of the vertex. It is well known that such a metric defines a complex structure on the surface. Hence we obtain a map from the set of triangulations $\Theta_h$ of the surface $M_h$ to the moduli space of complex structures on $M_h$: It is interesting to consider the image of the subset $\Theta_h^0 \subset \Theta_h$ of minimal triangulations of the surface under the map ${\mathcal C}_h$.In general, there are many minimal triangulations on each surface of genus $h \geqslant 2$. However, there are genera $h$ for which the minimal triangulations are special. This special feature consists in the fact that the one-dimensional skeleton of such a triangulation is a complete graph. For $h \geqslant 2$ such triangulations are quite difficult to imagine, but they exist! However, not for every genus $h$. These special triangulations are easy to see in genera $0$ and $1$. If $h=0$, then a special triangulation corresponds to a tetrahedron and is obviously unique. If we replace each $2$-simplex of this triangulation by a spherical equilateral triangle of Gaussian curvature $1$ with angles $2\pi/3$, then we obtain a sphere of radius $1$. Note that the choice of the curvature of the metric on each $2$-simplex corresponds logically to the genus of the surface under consideration. The question of the image of the map ${\mathcal C}_0$ is meaningless, since there is a unique complex structure on $M_0$. In the case of $h=1$ the minimal triangulation is also unique, which is not so obvious, and its skeleton is also a complete graph, the graph $A_7$. If we replace each $2$-simplex by a flat equilateral triangle with side length $1$, then all triangles glue together to form a flat torus, since the valence of each vertex is $6$. Miraculously, the resulting flat torus is a flat $6$-gonal torus, that is, it can be obtained by gluing from a regular $6$-gon of an appropriate size (see Fig. 1 (b)). Thus, the map ${\mathcal C}_1$, as applied to the triangulation above, yields an extremal point, ${\mathbf e}^{(\pi/3)i}$, in the moduli space of complex tori. Note also that the flat metric obtained by the minimal triangulation turns out to be systolic extremal, that is, the so-called Loewner metric. If $h=2,3,4,5$, then there are no triangulations on $M_h$ that correspond to complete graphs. To obtain a full answer, we must perform simple calculations, which we leave to the reader. The result of these calculations is that triangulations corresponding to complete graphs can exist only if the genus of the surface belongs to one of the following two families:
$$
\begin{equation}
(F_0)\colon h=12k^2 \pm k, \quad \text{corresponds to the graph} \ A_{12k+3+(1 \pm 1)/2};
\end{equation}
\tag{19}
$$
and
$$
\begin{equation}
(F_1)\colon h=12k^2 \pm 7k+1,\quad \text{corresponds to the graph} \ A_{12k+7(1 \pm 1)/2},
\end{equation}
\tag{20}
$$
where $k=0,1,2,\dots$ . For $k=0$ we have already seen such triangulations on a sphere and a torus. With these formulae for potential genera, a simple check using the Jungermann–Ringel formula (see (11) or [35]) shows that all such triangulations actually exist and are minimal for any given genus $h$. Therefore, the next surface on which special triangulations appear has genus $6$. If $h$ belongs to one of the families (19) or (20), then any special triangulation $\theta$ defines a natural hyperbolic metric $g_{\theta}$ on $M_h$, and therefore a complex structure on this surface. Let the triangulation $\theta$ correspond to the complete graph $A_s$, where $s$ and $h$ are related by the appropriate formula (19) or (20). The hyperbolic metric $g_{\theta}$ on $M_h$ is obtained by replacing each $2$-simplex $\theta$ by an equilateral hyperbolic triangle with angle $2\pi/(s-1)$. Note that the corresponding complex structure on $M_h$ coincides with the structure defined using Euclidean triangles, as done above. This takes us to the following quite natural problem. Problem 7.1. Let $h \geqslant 6$ belong to one of the families (19) or (20). (a) Calculate/estimate the cardinality of the set $\Theta_h^0$ of distinct minimal triangulations of $M_h$. (b) Describe the set ${\mathcal C}_h(\Theta_h^0) \subset {\mathcal M}_h$ of the corresponding complex structures. (c) Is it true that Here $\operatorname{sys}(g_{\theta})$ is the systole of the corresponding hyperbolic surface.8. $\operatorname{kw}$-complexityConsider the probem of controlling the number of simplices in simplicial polyhedra of fixed dimension satisfying prescribed homotopy conditions. An example of such a problem is the estimate of the combinatorial area of a group, which we considered in § 3. Another example, known for a long time, is the problem of an estimate for the minimum number of simplices of a given dimension in a polyhedron homotopically equivalent to $\mathbb{R}P^m$. The reader interested in the recent progress in this direction is referred to [1]. In general, only the number of simplices of the largest and smallest dimensions can be more or less successfully controlled. Let $G$ be a group, and set where $P$ is a $2$-polyhedron with fundamental group $G$. Recall that $s_k(P)$ denotes the number of $k$-simplices $P$. It turns out that this combinatorial invariant of the group has a transparent topological meaning. We briefly give the main definitions, referring the reader to [4] for more details.Topological spaces considered below are assumed to be path connected and locally contractible. It is not a great simplification to consider only $\operatorname{CW}$-complexes. Let $X$ be a topological space, and let $\mathcal{U}=\{U_{\alpha}\}_{\alpha \in A}$ be an open covering of $X$. A covering $\mathcal{U}$ is called good if all non-empty intersections
$$
\begin{equation*}
U_{\alpha_1} \cap U_{\alpha_2} \cap \cdots \cap U_{\alpha_n}
\end{equation*}
\notag
$$
are contractible for all natural $n$. Following Karoubi and Weibel [37], we denote by $\operatorname{ss}(X)$ the smallest possible cardinality of a good covering of $X$ (the smallest size). We are interested only in spaces $X$ for which $\operatorname{ss}(X)$ is finite. Such a space $X$ is homotopy equivalent to a finite polyhedron, namely, the nerve of a good covering (see [33]). Finally, we define the following homotopy invariant, called in [37] the covering type of $X$:
$$
\begin{equation*}
\operatorname{ct}(X):=\min_{Y \sim X} \operatorname{ss}(Y),
\end{equation*}
\notag
$$
where the minimum is taken over the spaces $Y$ homotopy equivalent to $X$. We note that the values of $\operatorname{ct}(X)$ and $\operatorname{ss}(X)$ can differ greatly even in the simplest case of dimension $1$, that is, on graphs. The reader will find a number of examples in [37]. Remark 8.1. The idea of good coverings and their use is not new and goes back to the classical works of Leray [43] and Weil [64]. Good coverings are convenient for the investigation of a number of problems in algebraic topology. The reader might think that the covering type is similar to the Lusternik–Schnirelmann category. This similarity is only superficial, and the two homotopy invariants are very different. For example, if $X$ is an $m$-dimensional $\operatorname{CW}$-complex, then its category does not exceed $m+1$, while its covering type can be arbitrarily large. Following [4], we define the $\operatorname{kw}$-complexity of $G$ by
$$
\begin{equation*}
\operatorname{kw}(G):=\min_{\pi_1(X)\simeq G}\operatorname{ct}(X).
\end{equation*}
\notag
$$
Here the minimum is taken over the spaces $X$ with fundamental group $G$. The following theorem was proved in [6]. The explicit computations of $\operatorname{kw}$-complexity known to date are presented in the following two examples. Example 8.3. In contrast to the $\kappa$-area, which vanishes at all free groups, the computation of $\operatorname{kw}(\mathbb{F}_n)$ presents the first non-trivial example. The following formula is based on the explicit computation [37] of the covering type of a wedge of $n$ circles: where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Example 8.4. As in the case of $\kappa$-area, it is possible to calculate the $\operatorname{kw}$-complexity explicitly for groups of surfaces. We need the values of the chromatic number of surfaces in its dependence on the genus. Note that the chromatic number of a surface coincides with the number of vertices in its minimal triangulation (see [55]). As in Example 4.5, we denote the genus of the surface by $h$ with sign $\pm$ depending on orientability. The chromatic number of a surface is calculated by the formulae and The $\operatorname{kw}$-complexity of surface groups was calculated in [15]. Namely, for all surface groups except $\pi(2)^+$ we have and In the same paper Borghini and Minian showed that for all groups except $\pi(2)^+$, the minimal polyhedron is the surface itself, equipped with some minimal triangulation. As always, a sphere with two handles differs from all other surfaces: the $\operatorname{kw}$-minimal polyhedron of its fundamental group is not homeomorphic to the surface itself! Another peculiarity of the group $\pi(2)^+$ is as follows: while for all other surface groups the $\operatorname{kw}$-minimal polyhedron is minimal in the sense of $\kappa$-area, for $\pi(2)^+$ this is not the case, as explained below.There are different minimal triangulations of the surface $M_2^+$. One of them can be obtained starting from the minimal triangulation of a torus (see Fig. 1 (b)). Take two copies of the torus equipped with a minimal triangulation and remove from each torus one rhombus formed by two adjacent triangles (the choice of the rhombus is irrelevant). Glue the resulting holes along the edges with a rotation of $90^{\circ}$, that is, an acute-angled vertex of one rhombus is glued to an obtuse-angled vertex of the other. This yields a sphere with two handles and a well-defined triangulation on it. The triangulation constructed contains $24$ triangles, hence it is minimal. This triangulation has combinatorial diameter $2$. Note that gluing the holed tori along the boundary of the cut out rhombi by means of the identity map produces a pseudotriangulation. There is at least one more triangulation, found by Jungermann and Ringel [35], which has combinatorial diameter $3$. Since there are no minimal triangulations of this diameter on other surfaces, we describe this special triangulation briefly, following [35]. To do this, it is convenient to use the dual good covering. Figure 2 shows a good covering of the sphere $S^2$ by ten closed sets. The interior of the large circle in the figure corresponds to the upper hemisphere, and the exterior, the region at number 10, to the lower hemisphere. The multiplicity of this covering is three. Next, four small discs are cut out from the large disc marked by a dotted line. The boundaries of the resulting holes are glued crosswise, as shown in the figure, resulting in a sphere with two handles. We emphasize that gluing along small circles should be done so that the multiplicity of the covering obtained on $M^+_2$ does not increase and remains equal to three, and so that the covering itself remains good. It is easy to see that such a gluing is always possible. The triangulation of $M^+_2$ is the nerve of the good covering constructed. Vertices of this triangulation correspond to regions of the covering. If vertex $A$ corresponds to region $1$ and vertex $D$ corresponds to region $10$, then these vertices lie at a distance of $3$ from each other. This is shown in Fig. 3, where these vertices are connected by a path of three edges passing through some vertices $B$ and $C$ of the triangulation constructed. Any minimal triangulation of $M^+_2$ contains ten vertices. However, the Jungermann–Ringel triangulation allows one to reduce the $\operatorname{kw}$-complexity of the group $\pi(2)^+$ as described in the following remark. Remark 8.5. If $P$ is a $\operatorname{kw}$-minimal polyhedron of some group $G$, then the combinatorial distance between any two of its vertices is at most $2$. Indeed, if we assume that this is not the case and $A$ and $D$ are two vertices of $P$ such that the distance between them is $3$, then there is a shortest path from $A$ to $D$ consisting of three consecutive edges of $P$: see Fig. 3 (a). Take a new polyhedron $P_1$ obtained from $P$ by gluing the vertices $A$ and $D$ and filling in the $1$-cycle formed by the edges in question with one triangle (a $2$-simplex); see Fig. 3 (b). The resulting polyhedron $P_1$ is homotopy equivalent to $P$, but the number of its vertices is strictly smaller, which means that $P$ is not $\operatorname{kw}$-minimal. Taking the Jungermann–Ringel triangulation as the polyhedron $P$, we apply the above construction to reduce the number of vertices. This yields a $\operatorname{kw}$-minimal polyhedron of $\pi(2)^+$ that is not homeomorphic to a sphere with two handles. Since $\operatorname{kw}$-complexity is a relatively new invariant, its properties are not well understood, and there are a number of open questions related to it. The next proposition, proved in [4], and the discussion that follows give some insight into the behaviour of this invariant. Proposition 8.6. Let $G_i$, $i=1,2$, be two groups. Then the following properties hold. $1^{\circ}$. $\operatorname{kw}(G_1 \times G_2) \leqslant \operatorname{kw}(G_1) \operatorname{kw}(G_2)$. $2^{\circ}$. $\operatorname{kw}(G_1 \ast G_2) \leqslant \operatorname{kw}(G_1)+\operatorname{kw}(G_2)-3+a$, where $a=0$ if none of the groups is free, and $a=1$ otherwise. $3^{\circ}$. If $G$ is a group and $H < G$ is a subgroup of index $k$, then the inequality $\operatorname{kw}(H) \leqslant k\operatorname{kw}(G)$ holds. $4^{\circ}$. Let $\operatorname{kw}(G)=n$. Then the Betti numbers of $G$ satisfy the inequalities It seems that inequality $1^{\circ}$ in Proposition 8.6 is never sharp if both groups are non-trivial. The simplest example: $G_1=G_2=\mathbb{Z}$. In this case we have on the other hand
$$
\begin{equation*}
\operatorname{kw}(G_1\times G_2)=\operatorname{kw}(\pi(1)^+)=7.
\end{equation*}
\notag
$$
At the same time, inequality $2^{\circ}$ cannot be improved in general. Take an integer $k \geqslant 3$, and let $n=(k-1)(k-2)/2$. Let $G_1=\mathbb{F}_n$ and $G_2=\mathbb{Z}$. Then, as Example 8.3 shows,
$$
\begin{equation*}
\operatorname{kw}(G_1)=k\quad\text{and} \quad \operatorname{kw}(G_2)=3,
\end{equation*}
\notag
$$
on the other hand,
$$
\begin{equation*}
\operatorname{kw}(G_1\ast G_2)=\operatorname{kw}(\mathbb{F}_{n+1})=k+1.
\end{equation*}
\notag
$$
In § 4 we saw that the $\kappa$-area of a group is not very sensitive to free factors of the group (see Conjecture 4.7 and Theorem 4.8). In contrast, $\operatorname{kw}$-complexity is non-trivial for free groups and, in general, changes when free factors are added. However, this change is rather mysterious and poorly controlled in general. For example, for surface groups we have
$$
\begin{equation*}
\operatorname{kw}(\mathbb{Z}_2 \ast \mathbb{Z})= \operatorname{kw}(\pi(1)^-)+1=7
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\operatorname{kw}(\pi(1)^+ \ast \mathbb{Z})=\operatorname{kw}(\pi(1)^+)+1=8,
\end{equation*}
\notag
$$
on the other hand
$$
\begin{equation*}
\operatorname{kw}(\pi(2)^+ \ast \mathbb{Z})=\operatorname{kw}(\pi(2)^+).
\end{equation*}
\notag
$$
This leads us to the following conjecture. Here the first line corresponds to those genera for which minimal triangulations are associated with complete graphs (see (19) and (20)). Similar formulae can be written in the case of groups of non-orientable surfaces. Although no other exact calculations of $\operatorname{kw}$-complexity are known to date, it is often possible to estimate this complexity in terms of the algebraic characteristics of the group under consideration. A number of such results for Artin, Coxeter, and some other groups can be found in [4]. To keep the text simple, we present only a couple of examples. We define the following function of two positive integer arguments:
$$
\begin{equation*}
k(n,m)=\begin{cases} \dfrac{\sqrt{8n+1}+3}{2} & \text{for} \ m \leqslant \dfrac{n(\sqrt{8n+1}-3)}{6}\,, \vphantom{\biggr\}} \\ \sqrt[3]{6m}+2 & \text{in the other cases}. \end{cases}
\end{equation*}
\notag
$$
Corollary 8.9. Let $\mathcal{A}_n$ be a free Abelian group of rank $n$. Then the following inequalities hold: If $n=2$, then inequalities (22) yield $4 \leqslant \operatorname{kw}(\mathcal{A}_2) \leqslant 7$. Since $\mathcal{A}_2 \simeq \pi(1)^+$, Example 8.4 implies that $\operatorname{kw}(\mathcal{A}_2)=7$. For the group $\mathcal{A}_3$, (22) yields $5 \leqslant \operatorname{kw}(\mathcal{A}_3) \leqslant 13$, and the precise value of $\operatorname{kw}(\mathcal{A}_3)$ is unknown. Remark 8.10. A quick glance at (21) and (22) shows that the lower and upper bounds differ in order, which can lead to a strong overestimation or underestimation of the corresponding bounds. For example, if we set $m=0$ in (21), which corresponds to the case of a free group of rank $n$, then the lower bound has order $\sqrt{n}$ . In this case it coincides with the explicit calculation in Example 8.3. The upper bound has order $n$ and is therefore greatly overestimated. Other examples can also be given. Of course, this can be attributed to the shortcomings of the methods used to prove (21) and (22). Nevertheless, we observe the following. The behaviour of the number of vertices in minimal triangulations is much less stable than the behaviour of the number of simplices of maximum dimension (in our case the dimension is always equal to $2$). For example, it is well known [8] that in any triangulation of $\mathbb{R} P^m$ the number of $m$-simplices is at least $2^{m+1}$. Recently, in [1] triangulations of $\mathbb{R} P^m$ were found with number of vertices growing subexponentially in $m$. To conclude this section, we give an estimate of the $\operatorname{kw}$-complexity for finite Abelian groups (see details in [4]). It is useful to compare the following results with Theorem 6.7 and Corollary 6.8 for $\kappa$-area. Theorem 8.11. For any finite Abelian group $G$ the following double inequality holds: where $s$ is the number of invariant factors of $G$. This implies directly the following statement. In the last two results we see again a discrepancy between the orders of growth of the upper and lower bounds. The question is actually not as trivial as it can seem. It is quite possible that not only the order $m$ of the group as such affects the behaviour of $\operatorname{kw}(\mathbb{Z}_m)$, but also the length of its dyadic expansion (see the construction of the Möbius telescope in [4]). 9. Relationship of $\operatorname{kw}$-complexity with systolic and combinatorial areasTheorem [4] below relates the systolic area of a group to its $\operatorname{kw}$-complexity. Theorem 9.1. Let $G$ be a group. Then Furthermore, if $G$ has free index zero, then Remark 9.2. The condition of zero free index in the second part of the theorem is necessary. As the following example shows, if this condition is dropped, then the second inequality does not hold. Consider an arbitrary non-free group $G$. Then The first inequality of Theorem 9.1 is proved by constructing a special metric on some $\operatorname{kw}$-minimal polyhedron of the group $G$. This metric is useful in many constructions of systolic and asymptotic geometry; we return to it and describe it in detail in the proof of Theorem 13.1. The second inequality of the theorem is proved by choosing an $\varepsilon$-optimal Riemannian polyhedron for $\sigma(G)$ (see [4] for details). In Theorem 9.1 we see again different growth rates of the upper and lower bounds of $\sigma(G)$. This leads us to a question similar to Question 8.13 in the previous section: Question 9.3. Is there a function $\psi(t)$, $t > 0$, such that for any group $G$ of free index zero the equivalence holds? Let us compare two combinatorial invariants of groups, $\operatorname{kw}$-complexity and $\kappa$-area. The following proposition was proved in [4]: Proposition 9.4. For any group $G$ the inequality holds. Furthermore, if $G$ has free index zero, then Although the proof of this proposition is much simpler than that of Theorem 9.1 (see [4]), the structure of these statements is strikingly identical. In replacing $\operatorname{kw}(G)$ by $\sigma(G)$ we only need to change constants. We do not have a convincing explanation for this similarity, but we note that for groups of free index zero the first inequality of Theorem 9.1 can be deduced, although with a weaker constant, from Proposition 9.4 and Theorem 6.1, but the second inequality of Theorem 9.1 cannot be obtained by such a derivation. In conclusion, we note that the obvious analogue of Question 9.3 for the invariants $\operatorname{kw}(G)$ and $\kappa(G)$ also remains open. 10. Volume entropy of polyhedraIn this and the following sections we define and discuss another group invariant of differential geometric nature, now called volume entropy. As applied to manifolds, this same invariant is called asymptotic volume (see [2]). We begin with the definition of the volume entropy of Riemannian polyhedra. Let $G$ be a finitely generated group; at this point we do not restrict the number of relations. Consider an $m$-dimensional Riemannian polyhedron $(\operatorname{Z},\operatorname{g})$, and assume that $G$ acts on $\operatorname{Z}$ freely and isometrically with respect to the distance $\rho_{\rm g}$. It will not be too restrictive to assume that this action is simplicial and that the quotient space $\operatorname{Z}/G$ is a finite pseudosimplicial complex. The latter condition ensures that the action is completely discontinuous and cocompact. Note that, although the action of $G$ on $\operatorname{Z}$ is simplicial, it would be too restrictive to assume that the quotient space $\operatorname{Z}/G$ is simplicial. This is shown by the simple examples where $\operatorname{Z}$ is the Cayley graph of the group $G$, or $\operatorname{Z}$ is the simplicial complex which is the universal covering of a semisimplicial realisation of $K(G,1)$ (see [34]). We choose a point $q \in \operatorname{Z}$, and let
$$
\begin{equation*}
B(q,R;\operatorname{g})=\{p \in \operatorname{Z} \mid \rho_{\operatorname{g}}(p,q) \leqslant R\}
\end{equation*}
\notag
$$
be the geodesic ball of radius $R$ with centre $q$. It is well known that the limit in the following definition exists and does not depend on $q$: Definition 10.1. The quantity Example 10.2. Consider a finitely generated group $G$, and let $S$ be a finite family of its generators, which we always assume to be symmetric, that is, $S=S^{-1}$. We let $\operatorname{Z}=C_S$ be the right Cayley graph of $G$, and consider the left action of $G$ on $\operatorname{Z}$. The simplicial polyhedron $\operatorname{Z}$ is one-dimensional, so specifying a Riemannian metric on $\operatorname{Z}$ is equivalent to specifying the edge lengths of this graph. The invariance requirement imposed on the metric means that we need to specify only the lengths of the edges incident to the identity of the group and corresponding to the generators of $S$, that is, the set of positive numbers corresponding to pairs $\{a,a^{-1}\} \subset S$ of different generators of our family. This fully defines a $G$-invariant metric $\operatorname{g}=g_S$ on $C_S$, turning $C_S$ into a so-called weighted graph. Applying (23) to this situation, we obtain the relative algebraic entropy $\omega(G,g_S)$ of $G$. In group theory the above construction is usually simplified by considering only equilateral metrics $g^{\ast}_S$ on the Cayley graph $C_S$. The latter means that the length of each edge of $C_S$ is taken to be $1$ (see, for example, [31] or [32]). With this simplification, the metric $g^{\ast}_S$ on $C_S$ (and therefore on $G$) becomes the so-called word metric in the alphabet $S$. The reader will find more details and a significant number of illustrative examples in [31]. In conclusion, we note that using only word metrics is not a serious limitation. Here we extend the range of metrics under consideration on $C_S$ to arbitrary weighted metrics in order to include the algebraic entropy of groups in the general setup of volume entropy (see Definition 10.5 and Remark 10.6 below). We recall the following definitions.
We end our discussion of group growth here, since this subject is of very minor relevance to the subject treated below. All groups under consideration in what follows must have exponential growth. Again, the interested readers are referred to the excellent book [31], a significant part of which is devoted specifically to growth. The following example is key to the further exposition. Example 10.3. Consider a finite Riemannian polyhedron $(X,g)$ of dimension $m$, and let $\operatorname{Z}=\widehat{X}$ be the universal covering of $X$, $\operatorname{g}=\widehat{g}$ be the pullback of the metric $g$, and let $G=\pi_1(X)$. Applying the general statement of Definition 10.1 to this situation we obtain the volume entropy of the Riemannian polyhedron $(X,g)$, which we denote by $\omega(X,g)$. As a special case of this example, consider a closed Riemannian manifold $(M,g)$ of dimension $m$. We can choose some smooth triangulation on it to fit the framework of Example 10.3, although this is not necessary. Applying the scheme of Example 10.3 we obtain the volume entropy $\omega(M,g)$ of the Riemannian manifold $(M,g)$. The basic idea underlying the definition of volume entropy is contained in Manning’s paper [47]. The following result in it explains the term ‘entropy’ used for invariants of this type. Theorem 10.4. Let $(M,g)$ be a closed Riemannian manifold and let $h(g)$ be the topological entropy of the geodesic flow of the metric $g$. Then the inequality holds. Moreover, if the sectional curvature of the metric is nowhere positive, $k(g) \leqslant 0$, then (24) turns to equality. What happens when the metric on the Riemannian polyhedron $(\operatorname{Z},\operatorname{g})$ varies? For homotheties $\operatorname{g} \leftrightarrow t^2 \operatorname{g}$ we obviously have
$$
\begin{equation*}
\omega(\operatorname{Z},G;t^2\operatorname{g})= \frac{\omega(\operatorname{Z},G;\operatorname{g})}{t}\,.
\end{equation*}
\notag
$$
Here we introduce the coefficient $t^2$ since $\operatorname{g}$ always denotes a quadratic form, lengths are naturally multiplied by $t$, and volumes by $t^m$, $m=\dim \operatorname{Z}$. Definition 10.5. $1^{\circ}$. The minimum volume entropy of a $G$-action on an $m$-dimensional polyhedron $\operatorname{Z}$ is defined by where $\operatorname{g}$ ranges over the set of $G$-invariant metrics on $\operatorname{Z}$. The volume $\operatorname{Vol}(\operatorname{Z}/G)$ is well defined, since in the interior of each $m$-simplex the metric $\operatorname{g}$ is pushed forward uniquely from $\operatorname{Z}$ to the quotient space $\operatorname{Z}/G$. $2^{\circ}$. The algebraic entropy of a finitely generated group $G$ is the quantity
$$
\begin{equation}
\mathrm{ent}_{\rm alg}(G)=\inf_{S,g_S} \omega(G,g_S)l_{g_S}(S),
\end{equation}
\tag{25}
$$
where $l_{g_S}(S)$ is the total length of all generators of the system $S$ in the metric $g_S$. $3^{\circ}$. The minimum volume entropy of an $m$-dimensional finite simplicial polyhedron $X$ is the quantity where $g$ ranges over the set of simplicial metrics on $X$.Remark 10.6. In group theory algebraic entropy is usually understood as a simpler version of minimisation in comparison to (25): where $S$ ranges over all possible systems of generators of the group $G$, and $g^{\ast}_S$ is the equilateral, or word, metric associated with the system $S$. In other words, the total length of the system of generators, even with respect to the word metric, is not taken into account. Since the main part of the results on the algebraic entropy of groups was obtained precisely within the framework of this simplified definition, we refer to the quantity introduced in (26) as the weak algebraic entropy of the group $G$. For some reasons that we do not discuss here (see [6]), in dimensions $m > 1$ it is more convenient to work with the minimum entropy raised to the power equal to the dimension of the space. Thus, we will use the notation where $m=\dim X$, leaving the old notation $\mathrm{ent}_{\rm alg}(G)$ for the algebraic entropy of the group $G$.Example 10.7. For a free group of rank $n$ the equality holds. Apparently, this calculation was first carried out in [28] (see also [31] and [32]). Note that the quantity above differs from the one in [31] by a factor of $2n$. This is because part $2^{\circ}$ of Definition 10.5 minimizes $\omega(G,g_S)l_{g_S}(S)$. As with the other invariants encountered above, computing $\mathrm{ent}_{\rm alg}$ and $\mathrm{ent}_{\rm alg}^{\ast}$ explicitly is usually difficult. Nevertheless, in a number of interesting cases it is possible to find quite effective lower bounds for algebraic entropy. The reader will find many examples in [31] and [32]. Over the past 20 years significant progress has been made in the study of weak algebraic entropy, and it has been calculated in a number of cases. Here are some examples. Example 10.8. $1^{\circ}$. Groups of the form $G=\mathbb{F}_n\ast \mathbb{Z}_p$, where $p$ is a prime number, were considered in [61]. $2^{\circ}$. The weak algebraic entropy of free products of the form $\mathbb{Z}_2 \ast \mathbb{Z}_{p^k}$, where $p$ is a prime number, was studied in [62]. In the same paper the entropy of the groups $\mathbb{Z}_2 \ast \mathbb{Z}$ and $\mathbb{Z}_3 \ast \mathbb{Z}_3$, as well as of the Baumslag–Solitar groups $B(n,n)$, was calculated. The special cases $\mathbb{Z}_2 \ast \mathbb{Z}_3$ and $\mathbb{Z}_2 \ast \mathbb{Z}_4$ were previously considered in [46]. The calculations in the above cases yield the precise value of the invariant $\mathrm{ent}_{\rm alg}^{\ast}(\operatorname{PSL}(2,\mathbb{Z}))$. Using a central extension, this allows us to conclude that
$$
\begin{equation*}
\mathrm{ent}_{\rm alg}^{\ast}(\operatorname{SL}(2,\mathbb{Z}))= \mathrm{ent}_{\rm alg}^{\ast}(\operatorname{PSL}(2,\mathbb{Z}))= \frac{\ln 2}{2}\,.
\end{equation*}
\notag
$$
All precise values of the entropy, as well as the sets of generators on which these values are realised, can be found in the works mentioned above. Sometimes, these quantities are expressed in terms of zeros of rather complicated polynomials, so we do not present them. $3^{\circ}$. In [19] the weak algebraic entropy of the group $\operatorname{PGL}(2,\mathbb{Z})$ was calculated, among other things. This example deserves special attention, see the remark below. $4^{\circ}$. Finally, we note [20], where it was shown that for an odd prime $p$,
$$
\begin{equation*}
\mathrm{ent}_{\rm alg}^{\ast}(B(1,p))= \mathrm{ent}_{\rm alg}^{\ast}(\mathcal{L}_p).
\end{equation*}
\notag
$$
Here $B(1,p)$ is the Baumslag–Solitar group, and $\mathcal{L}_p=\mathbb{Z}_p \wr \mathbb{Z}$ is the so-called lamplighter group. For $p=2$ this equality is false (see details in [20]). In each of the above cases the value of weak algebraic entropy is attained at some (usually canonical) generating set. The group $\operatorname{PGL}(2,\mathbb{Z})$, considered in [19], is of interest in this connection. Here $\mathrm{ent}_{\rm alg}^{\ast}$ is attained not at the smallest generating set of two generators, but at the set of three Coxeter generators. Although in this example the system of three generators provides the least value of entropy, even if the total length of the system is taken into account, it suggests that the algebraic entropies $\mathrm{ent}_{\rm alg}^{\ast}$ and $\mathrm{ent}_{\rm alg}$ can differ significantly. So far, this issue has not been studied at all. If a finitely generated group $G$ has exponential growth, then the inequality $\omega(G,g_S) > 0$ holds for whatever the system of generators $S$ and the metric $g_S$ associated with it. It is not at all clear whether the algebraic entropy must be positive, that is, $\mathrm{ent}_{\rm alg}(G) > 0$, in this case. This problem was first noted in the Gromov’s book [28]. A fairly complete discussion of this topic for weak algebraic entropy is available in [32], where the question of when exponential growth actually implies the positivity of weak algebraic entropy is also considered. Among the classes of groups with positive weak algebraic entropy considered in [32], we note free products with amalgamation, $\operatorname{HNN}$-extensions, and one-relator groups. The reader will find details and effective lower bounds for algebraic entropy in these cases in [32]. It turns out that exponential growth is not at all sufficient for algebraic entropy to be positive. Wilson [65] constructed an example of such a finitely generated group: Theorem 10.9. There exists a $2$-generated group $G$ containing non-Abelian free subgroups and with a sequence of generating systems $\{S_n\}_{n=1}^{\infty}$ satisfying the following condition: Each system $S_n$ contains two generators, one of order 2 and the other of order 3. Here, as above, $g^{\ast}_{S_n}$ is the equilateral metric associated with the system of generators $S_n$. This is a rather subtle result since the group $\mathbb{Z}_2\ast\mathbb{Z}_3\simeq \operatorname{PSL}(2,\mathbb{Z})$ has a positive algebraic entropy (see Example 10.8). In connection with Wilson’s result we give the following definition. Definition 10.10. A finitely generated group $G$ has uniformly exponential growth of order at least $a>0$ if $\mathrm{ent}_{\rm alg}^{\ast}(G) \geqslant a$. The Wilson group has an infinite number of relations. The following problem is well known and remains wide open. Problem 10.11. Are there finitely presented groups of exponential but not uniformly exponential growth? We conclude this section with a problem close to Conjectures 3.3 and 11.2 on the existence of minimal polyhedra for systolic area and volume entropy, respectively. Wilson’s theorem shows that the algebraic entropy of a finitely generated group may not be attained on any system of generators. The presence of an infinite number of relations does not play a significant role here. Even before Wilson’s result appeared, Sambusetti [57] had shown that if a group $G_1$ is non-Hopfian and $G_2$ is a non-trivial group, then the following inequality holds for the group $G=G_1\ast G_2$:
$$
\begin{equation*}
\mathrm{ent}^{\ast}(G) < \omega(G,g^{\ast}_S) \quad \text{for any system of generators } S.
\end{equation*}
\notag
$$
The simplest example of a group satisfying this condition is $G=B(2,3)\ast \mathbb{Z}_2$, where $B(2,3)$ is the simplest non-Hopfian Baumslag–Solitar group [9]. Problem 10.12 ([32]). Describe finitely generated groups of uniformly exponential and attainable growth. Here uniformly exponential growth is attainable if there exists a system of generators $S$ such that
$$
\begin{equation*}
\mathrm{ent}_{\rm alg}^{\ast}(G)=\omega(G,g^{\ast}_S) > 0.
\end{equation*}
\notag
$$
All groups in Example 10.8 are groups of uniformly exponential and attainable growth. Despite the fact that the stock of examples is quite large, the problem above seems to be little studied. A similar problem can be stated for the algebraic entropy $\mathrm{ent}_{\rm alg}$, but we have even less information on it. 11. Volume entropy of non-free finitely presented groupsIn the rest of this paper we consider only finitely presented groups. Up to the last section, $G$ denotes a non-free group. The basic definitions and constructions follow [4] and [51]. Definition 11.1. The volume entropy of a group $G$ is the quantity where $X$ ranges over all two-dimensional finite simplicial polyhedra with fundamental group $G$. Since a free group is a one-dimensional object, the formal application of Definition 11.1 to a free group yields $0$. We consider a modification of this definition for free groups in the last section (see § 15.3). Just as in the case of systolic area, the question of the existence of a minimal polyhedron remains fully open (see Conjecture 3.3): Conjecture 11.2. For any non-free group $G$ there exists an entropy minimal polyhedron, that is, a $2$-polyhedron $P$ such that $\pi_1(P)=G$ and $\mathrm{Ent}(P)=\mathrm{Ent}(G)$. As in the case of systolic area, there are few examples of the explicit calculation of volume entropy in the case where it is positive. Example 11.3. Consider a surface group $\pi(h)^+$. For simplicity assume that the surface is orientable. Then Indeed, if $X$ is a $2$-polyhedron with fundamental group $\pi_1(X)=\pi(h)^+$, then applying the arguments from [16] we can show that where $N \subset X$ is an embedded surface realising the fundamental class $[M_h] \in H_2(X,\mathbb{Z})$. The genus of $N$ is generally greater than $h$, so in the last estimate we use the relative volume entropy $\mathrm{Ent}_{\ast}$ (see [6]). This implies that The following proposition, proved in [51], describes some general properties of volume entropy. Although the volume entropy and systolic area of a group are quite different in nature, this proposition is identical to Proposition 3.5, describing the properties of the systolic area. Proposition 11.4. $1^{\circ}$. Let $G_i$, $i=1,2$, be two groups. Then In particular, for any group $G$ and any natural integer $n$ $2^{\circ}$. Let $G$ be a group and $H < G$ be a subgroup of index $k$. Then As in the case of systolic area, Proposition 11.4 leads to a conjecture similar to Conjecture 3.7. Example 11.6. Consider the group $\operatorname{PSL}(2,\mathbb{Z}) \simeq \mathbb Z_2 \ast \mathbb Z_3$. From Proposition 11.4 we immediately obtain that $\mathrm{Ent}\bigl(\operatorname{PSL}(2,\mathbb{Z})\bigr)=0$. Thus, we see that the positivity of algebraic entropy does not means that volume entropy is positive too. Unlike systolic area, which is positive for any non-free group, volume entropy is often zero even for groups of uniformly exponential growth, and Example 11.6 is not at all exotic. Following [51], in the next section we define a fairly wide class of groups with zero volume entropy that contains numerous examples of known groups. 12. Soft groupsFirst, we recall the definition of the graph product of a system of groups. Of the various equivalent forms of this definition, we follow the one presented in [33], where the reader will find numerous examples. Multigraphs are allowed as directed graphs in [33], but we consider only graphs in the strict sense, for geometric reasons. Sometimes (as, for example, for the graph product representation of an $\operatorname{HNN}$-extension), this leads to a slight modification of the construction. Consider a finite connected directed graph $\Gamma$ with vertex set $V$ and edge set $E$. Suppose that to each vertex $v \in V$ a group $G_v$ is assigned, and to each edge $[v,v'] \in E$ a homomorphism $\varphi_{[vv']}\colon G_v \to G_{v'}$ is also assigned. Here $v$ is the outgoing vertex and $v'$ is the incoming vertex of the edge $[v,v']$. The family of groups and homomorphisms
$$
\begin{equation}
\Gamma(\mathcal{G})=\{G_v \}_{v\in V}\sqcup \{\varphi_e\}_{e\in E}
\end{equation}
\tag{27}
$$
is called a graph of groups. Now replace each group $G_v$ by its classifying cellular space $K(G_v,1)$, and each edge $[v,v']$ by the cylinder $K(G_v,1)\times [0,1]$. Next, for each homomorphism $\varphi_{vv'}$ choose a cellular map inducing a homomorphism $\varphi_{[vv']}$ on the fundamental groups. Finally, using the mapping cylinder construction, glue together all the cylinders constructed in accordance with the maps $\{\Phi_e\}_{e \in E}$. The resulting complex is denoted by $K\Gamma(\mathcal{G})$. The following classical result goes back to Whitehead; the proof can be found in [33]: Theorem 12.1. Suppose that in the above construction all homomorphisms of the family $\{\varphi_e\}_{e\in E}$ are monomorphisms. Then $K\Gamma(\mathcal{G})$ is an Eilenberg–MacLane space $K(G,1)$, and the inclusions $K(G_v,1) \hookrightarrow K\Gamma(\mathcal{G})$ induce monomorphisms of the fundamental groups. Definition 12.2. Let $\Gamma$ be a graph and (27) be some family of groups and homomorphisms corresponding to this graph. Assume that all homomorphisms are monomorphisms. The graph product of family (27) is the fundamental group of the corresponding complex $K\Gamma(\mathcal{G})$: Example 12.3. Let $\Gamma$ have three vertices $\{a,b,c\}$ and two edges $\{[c,a],[c,b]\}$ oriented in the way indicated. Consider three groups $G_a$, $G_b$, and $G_c$, where $G_c < G_a$ and $G_c < G_b$. The homomorphisms $\varphi_{ca}$ and $\varphi_{cb}$ are inclusions. The corresponding graph product is the amalgamated free product: In particular, there are graph product decompositions of classical groups: Example 12.4. Let $G$ and $H$ be two groups, where $H$ is included in $G$. Consider the graph $\Gamma$ with three vertices and three edges forming a triangle: where the edges are oriented in the way indicated. As vertex groups we take $G_3=G$ and $G_1=G_2=H$, and we choose the edge homomorphisms as follows: $\varphi_{12}$ is the identity isomorphism, and $\varphi_{13}$ and $\varphi_{23}$ are some inclusions of $H$ into $G$. In this case the resulting graph product is an $\operatorname{HNN}$-extension In the special case where all vertex groups are equal to $\mathbb{Z}$ we have $G_i=\mathbb{Z}$, $i=1,2,3$. We choose edge homomorphisms as follows: $\varphi_{12}$ is the identity isomorphism, $\varphi_{13}$ is multiplication by $m$, and $\varphi_{23}$ is multiplication by $n$, where $m$ and $n$ are non-zero integers. The corresponding graph product is a Baumslag–Solitar group ($\operatorname{BS}$-group) [9]: Definition 12.5. A group $G$ is called a generalized Baumslag–Solitar group ($\operatorname{GBS}$-group) if it facttorizes into a graph product, where all vertex groups are $\mathbb{Z}$. A factorization of a group into a graph product is not unique. For example, a surface group of high genus can be factorized into graph products in different and quite dissimilar ways. To the author’s knowledge, the question of when two graph products define isomorphic groups has virtually not been explored. Definition 12.6 ([51]). A group $G$ is called soft if it admits a graph product factorization such that all vertex groups have polynomial growth. As we see from the examples above, this is a fairly large class of groups, closed under the free product. On the other hand the groups of orientable surfaces of genus greater than $1$ and non-orientable surfaces of genus greater than $2$ do not belong to this class. All $\operatorname{GBS}$-groups have geometric dimension $2$, while soft groups can have an arbitrary geometric dimension, including infinity. Note also that many groups among soft ones have uniformly exponential growth, for example, almost all classical $\operatorname{BS}$-groups. Theorem 12.7 ([51]). For any soft group $G$ 13. Volume entropy and combinatorial invariants of groupsWe say that a group $G$ is indecomposable if it cannot be factorized into a free product with non-trivial factors. Corollary 13.2. There is a positive constant $E$ such that the following inequality holds for any indecomposable group $G$ satisfying $\mathrm{Ent}(G) \geqslant E$: Since this result has not yet been mentioned in the literature, we provide a brief proof of it. Proof of Theorem 13.1. Let $\kappa(G)\,{=}\,k$, and consider some $\kappa$-minimal $2$-polyhedron $P$, that is, $\pi_1(P)=G$ and $s_2(P)=k$. We equip $P$ with the following Riemannian metric $g$. On each $2$-simplex $\tau^2 \subset P$ the metric $g$ is the metric of a sphere of radius $r=2/\pi$ in which $\tau^2$ is an equilateral spherical triangle with right angles. In this case each edge of $\tau^2$ has length $1$, and the area of $\tau^2$ is equal to $2/\pi$. Thus, the metrics defined on each $2$-simplex glue together into a global $\operatorname{PL}$-metric on $P$, which we also denote by $g$. The area of the Riemannian polyhedron $(P,g)$ is
Since $G$ is indecomposable, every $1$-simplex of $P$ is incident to some $2$-simplex, and the condition of $\kappa$-minimality for $P$ implies that every $1$-simplex is incident to at least two $2$-simplices. Thus, the $1$-skeleton $P^{(1)} \subset P$ is a graph all of whose edges have length $1$ in the metric $g$. Consider the embedding In [4], Lemma 4.1, it was proved that this embedding is globally isometric on the vertex set.Now consider the universal covering $\widehat{P} \to P$ of $P$, and let $\widehat{g}$ be the pullback of the metric $g$. Obviously, $\widehat{P}^{(1)}$ is a covering of $P^{(1)}$. Just as in the case of (31), the embedding
$$
\begin{equation}
(\widehat{P}^{(1)},\widehat{g}) \hookrightarrow (\widehat{P},\widehat{g})
\end{equation}
\tag{32}
$$
is globally isometric on the vertex set.
Let $m$ be the maximum valency of vertices in the graph $\widehat{P}^{(1)}$; it coincides with the maximum valency of vertices of $P^{(1)}$. If $q \in P^{(1)}$ is a vertex of valency $m$, then, since each of $m$ edges is incident to at least two $2$-simplices, the vertex $q$ is incident to $s \geqslant m$ $2$-simplices. This means that the link $L(q)$ of the vertex $q$ consists of $s$ edges, each of which is incident to at least two $2$-simplices. Simplices incident to edges of $L(q)$, but not incident to the vertex $q$ itself, are called ‘new’. None of the new $2$-simplices can have all three edges in $L(q)$, for otherwise there is a tetrahedron contained in $P$, and this contradicts the minimality of $P$. Thus, there are at least $s/2$ new $2$-simplices. So we have already found $s+s/2$ different $2$-simplices in $P$, and therefore $s \leqslant (2/3)k$, which implies the inequality Now we choose some vertex $\widehat{q} \in \widehat{P}$ and consider the ball $B(\widehat{q},t;\widehat{g})$ of radius $t$ with centre $\widehat{q}$ in the metric $\widehat{g}$. Denoting by $|X|$ the cardinality of the set $X$, we obtain the obvious inequality
$$
\begin{equation*}
V(\widehat{q},t;\widehat{g})= \operatorname{Vol}\bigl(B(\widehat{q},t;\widehat{g})\bigr) \leqslant |B(\widehat{q},t;\widehat{g}) \cap \widehat{P}^{(0)}|\operatorname{Vol}(P,g).
\end{equation*}
\notag
$$
Since the embedding (32) is isometric on the vertex set and the maximum valency of vertices of $\widehat{P}^{(1)}$ does not exceed $m$, the number of vertices of $\widehat{P}$ at a distance of at most $t$ from $\widehat{q}$ does not exceed $m^t$. Taking (33) and (30) into account we obtain
$$
\begin{equation*}
V(\widehat{q},t;\widehat{g}) \leqslant \biggl(\frac{2}{3}\,k\biggr)^t k\, \frac{2}{\pi}=\frac{3}{\pi}\biggl(\frac{2}{3}\,k\biggr)^{t+1}.
\end{equation*}
\notag
$$
From this we conclude that
$$
\begin{equation*}
\omega(P,g) \leqslant \ln \biggl(\frac{2}{3}\,k\biggr).
\end{equation*}
\notag
$$
This, in turn, implies the estimate
$$
\begin{equation*}
\mathrm{Ent}(P,g) \leqslant \frac{3}{\pi}\biggl(\frac{2}{3}\,k\biggr) \biggl(\ln \biggl(\frac{2}{3}\,k\biggr)\biggr)^2,
\end{equation*}
\notag
$$
from which the required inequality (28) follows. $\Box$ To prove Corollary 13.2 consider the function which is strictly increasing if $t \geqslant e^2$ and is asymptotically inverse to the function The latter implies that there exists a constant $t_0 > e^2$ such that if $t \geqslant t_0$, then $f(g(t)) \leqslant 2t$, where $t_0$ is determined by $f(t)$ alone.Applying $f(t)$ to (28) and assuming that we obtain
$$
\begin{equation*}
\frac{\pi}{3}\,\frac{\mathrm{Ent}(G)}{(\ln(\mathrm{Ent}(G)))^2} \leqslant \frac{4}{3}\,\kappa(G).
\end{equation*}
\notag
$$
The corollary is proved. Remark 13.3. Note that the ‘mysterious’ function $t/(\ln t)^2$, which we have already encountered several times in § 6, reappears in the above proof. If no additional conditions, apart from indecomposability, are imposed on the group $G$, then the inequality $\kappa(G) \leqslant F(\mathrm{Ent}(G))$, opposite to (29), cannot be true for any function $F$. This takes us to the following problem. Problem 13.4. Find sufficient algebraic or topological conditions on a group $G$ so that the inequality $\kappa(G) \leqslant F(\mathrm{Ent}(G))$ holds for some suitable function $F$. For any group $G$ its volume entropy can be estimated from above in terms of the $\operatorname{kw}$-complexity: the following result was obtained in [4]: 14. Geometrically finite groups and volume entropy in the sense of Bregman and ClayAnother version of volume entropy was recently proposed by Bregman and Kley [17]. Formally, this type of entropy is defined following a scheme similar to the one used in § 11. However, in the situation considered below, this scheme cannot be applied to every finitely presented group. We begin with some definitions. Definition 14.1. A group $G$ is called geometrically finite (or a gf-group) if its classifying space $K(G,1)$ can be realised by a finite cell (and hence simplicial) complex. The geometric dimension of a geometrically finite group $G$, denoted by $\dim_{\rm g}(G)$, is the smallest possible dimension of its classifying space. Although gf-groups represent a relatively small part of all groups, they are a rather interesting and natural class of finitely presented groups. On the one hand it generalises the class of fundamental groups of aspherical manifolds. On the other hand it contains, for instance, many one-relator groups. More specifically, according to a well-known theorem of Lyndon, a one-relator group is geometrically finite if and only if the word defining the relation is not a non-trivial power of another word. All non-free one-relator gf-groups have geometric dimension $2$, and the complex constructed by attaching a $2$-cell to a wedge of circles by the relation is aspherical (see [25]). The class of gf-groups contains all $\operatorname{GBS}$-groups; they also have geometric dimension $2$. More generally, a graph product of gf-groups is a gf-group. Another subclass of gf-groups consists of groups acting freely and cocompactly on some $\text{Cat}(0)$-complete cubic complex of finite dimension, such as, for example, all right-angled Artin groups (see [23]). There are also other interesting subclasses of gf-groups, which we do not consider here. In conclusion of this short excursion to gf-groups, we note that the geometric dimension of a gf-group $G$ ‘almost always’ coincides with its cohomological dimension where $M$ ranges over all possible $G$-modules. For other interpretations of this dimension in terms of the length of the projective resolution of $\mathbb{Z}$ as a ${\mathbb{Z}}G$-module, see [18].The expression ‘almost always’ refers here to the dimension, rather than to the group, and means the following. According to Stallings [59] and Swan [60], a group $G$ has cohomological dimension $1$ if and only if it is free. Therefore, $\dim_{\rm c}(G)=\dim_{\rm g}(G)=1$ in this case. Since for any group $\dim_{\rm c}(G) \leqslant \dim_{\rm g}(G)$, it also follows that if $\dim_{\rm g}(G)=2$, then $\dim_{\rm c}(G)=\dim_{\rm g}(G)$. Finally, according to the well-known Eilenberg–Ganea theorem [26] (see also [18]), for any gf-group $G$ there exists a finite classifying space of dimension $\max\{\dim_{\rm c}(G),3\}$. The following question still remains unclear. Now we turn to entropy itself; the following definition is similar to Definition 11.1. Definition 14.3. Let $G$ be a gf-group of geometric dimension $m$. Its volume entropy is the quantity where $X$ ranges over finite aspherical simplicial polyhedra of dimension $m$ with fundamental group $G$. In this definition the geometric dimension of the group plays a key role, so the dimension is reflected in the notation. The following conjecture is quite analogous to Conjecture 11.2 formulated for the usual volume entropy of a group. Conjecture 14.4. For any gf-group $G$ of geometric dimension $m$ there exists an entropy minimal polyhedron, that is, an aspherical $m$-polyhedron $P$ such that As we will see in § 15, this conjecture is valid in dimension $m=1$. If $m > 2$, then for a gf-group $G$ the entropies $\mathrm{Ent}^{(m)}(G)$ and $\mathrm{Ent}(G)$ generally have nothing in common. Example 14.5 ([51]). Consider the group $G=\pi(2)^+ \ast(\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z})$. It is geometrically finite and has geometric dimension $m=3$, and its classifying space can be taken to be $P=M^+_2 \vee T^3$. From Proposition 14.7 below we obtain Here the equality $\mathrm{Ent}^{(3)}(\pi(2)^+)=0$ follows from dimensional considerations, and the equality $\mathrm{Ent}^{(3)}(\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z})=0$ is valid since this group has polynomial growth. On the other hand it was shown in [51] that $\mathrm{Ent}(G) > 0$. As we have seen, groups of geometric dimension two form a fairly diverse subfamily of gf-groups. In this case Definitions 11.1 and 14.3 ‘almost’ coincide. The only difference is that in the first case the infimum is taken over all $2$-polyhedra with fixed fundamental group, and in the second case only over aspherical $2$-polyhedra with this fundamental group. Consequently, for any gf-group $G$ of geometric dimension $2$ we have There is also an analogue of Proposition 11.4 for the entropy of gf-groups (see [51]): Proposition 14.7. $1^{\circ}$. Let $G_i$, $i=1,2$, be two gf-groups. If $m=\dim_{\rm g}(G_1)=\dim_{\rm g}(G_2) > 1$, then $2^{\circ}$. Let $G$ be a gf-group of dimension $\dim_{\rm g}(G)=m$ and $H < G$ be a subgroup of index $k$. Then In part $2^{\circ}$ of Proposition 14.7 entropy of dimension $m=\dim_{\rm g}(G)$ is taken on both sides of the inequality. Obviously, $\dim_{\rm g}(G) \geqslant \dim_{\rm g}(H)$, so even if this inequality is strict, the inequality in part $2^{\circ}$ of Proposition 14.7 remains true, since its left-hand side turns to zero. Note that the dimensions $\dim_{\rm g}(G)$ and $\dim_{\rm g}(H)$ coincide ‘almost always’. Since $G$ is a gf-group, it is torsion free, and by Serre’s well-known theorem [58] the cohomological dimensions of $G$ and $H$ coincide: $\dim_{\rm c}(G)=\dim_{\rm c}(H)$. Applying the Eilenberg–Ganea theorem [26] mentioned above, we obtain that if $\dim_{\rm g}(G) \ne 3$, then $\dim_{\rm g}(G)=\dim_{\rm g}(H)$. As above, a question similar to Problem 14.2 remains open. Problem 14.8. Is there a gf-group of geometric dimension $3$ that has a subgroup of finite index of geometric dimension $2$? The author does not know of any analogues of Theorems 13.1 and 13.5 for the entropy of gf-groups. If $\dim_{\rm g}(G)=m > 2$, then the invariants $\kappa(G)$ and $\operatorname{kw}(G)$ are clearly insufficient to obtain any estimates for $\mathrm{Ent}^{(m)}(G)$. Perhaps this can be done by invoking all ranks of the minimal finite projective resolution of $\mathbb{Z}$ as a ${\mathbb{Z}}G$-module, but this question has not been studied at all. If $\dim_{\rm g}(G)=2$, then it would seem that the existing proofs of Theorems 13.1 and 13.5 can be transferred directly to this case. However, these proofs appeal to $\kappa$- or $\operatorname{kw}$-minimal polyhedra, and it is absolutely unclear why such polyhedra should be aspherical. No analysis of this situation has been made yet. We conclude this section with the following assertion, which follows directly from Proposition 3.5 in § 3 and Theorem 3.23 in [5]. Theorem 14.9. Let $G$ be a word hyperbolic group of cohomological dimension $m \geqslant 3$, and let $H^m(G,\mathbb{R}) \ne 0$. Then 15. Free groupsIn this section we analyse what happens to the geometric and combinatorial invariants considered in our paper when we apply them to a free group. Some of the definitions given above require a slight dimensional adjustment. As we already noted in § 14, free groups are the only groups of cohomological and geometric dimension $1$. If $\mathbb{F}_n$ is a free group of rank $n$, then the set of finite one-dimensional simplicial complexes with fundamental group $\mathbb{F}_n$ consists of graphs of Euler characteristic $1-n$. This immediately transforms differential-geometric problems on free groups into the corresponding problems on graphs. For dimensional reasons, in place of area we will consider the total length of graphs. 15.1. The systolic length of graphsLet $(\Gamma,g)$ be a metric (or weighted) graph. As above, by the systole $\operatorname{sys}(\Gamma,g)$ we mean the length of the shortest non-contractible closed path on $\Gamma$. This characteristic of a graph was known under the name girth and had been studied (see [63], [49], and [27]) long before the term ‘systole’ came into general geometric use. Also see [13], containing an extensive bibliography. In what follows we nevertheless adhere to the term ‘systole’. Let $(\Gamma,g)$ be a metric graph, and let $L_g(\Gamma)$ denote its total length, that is, the sum of the lengths of all edges in the metric $g$. The systolic length of $\Gamma$ is defined by
$$
\begin{equation}
\sigma(\Gamma):=\inf_g \frac{L_g(\Gamma)}{\operatorname{sys}(\Gamma,g)}\,,
\end{equation}
\tag{34}
$$
where $g$ ranges over all possible metrics on $\Gamma$. Note that this definition is in line with the general definition (4) of the systolic volume of a simplicial polyhedron. Finally, the systolic length of a free group of rank $n$ is defined by where $\Gamma$ ranges over all graphs with fundamental group $\mathbb{F}_n$. Remark 15.1. In (34) and (35) we use all possible metrics on graphs to bring these definitions as close as possible to the ones we have already encountered: the definition of the systolic volume of a polyhedron and the definition of the systolic area of a group. It can be seen that the lower bound in (34) does not change if we use only metrics with rational values of lengths. Since the ratio in (34) is invariant under homotheties (dilations and contractions) of the metric, the value of this ratio for a rational metric is equal to its value for some metric with integer lengths of edges. By making an appropriate subdivision of edges, we arrive at a new graph with the same systolic ratio as in (34), but with combinatorial metric. Thus, in (35) we could minimize over combinatorial graphs alone, calculating their combinatorial length and systole. This point of view is better suited to the general logic of graph theory. Calculating the systolic length of a free group $\mathbb{F}_n$ as a function of $n$ turned out to be not as simple as it might seem a priori. We start with a result due to Erdös and Sachs [27], which was proved more than 60 years ago. Theorem 15.2. Given $d \geqslant 2$ and $l\geqslant 4$, set Then for $n \geqslant N(d,l)$ there exists a $d$-regular Hamiltonian graph $\Gamma$ with $2n$ vertices satisfying the inequality $\operatorname{sys}(\Gamma) \geqslant l$ with respect to the combinatorial metric. Recall that a graph is called $d$-regular if each of its vertices has valency $d$, and a graph is Hamiltonian if it has a simple closed cycle passing through each vertex. It follows directly from this theorem that if $n\geqslant 2$, then The above result of Erdös and Sachs was the first serious advance in the study of the systolic length of graphs. Since then, the estimates given have been improved many times. The systolic topic is quite popular in graph theory, and there is an extensive literature, where many results are obtained just in the logic of graph theory. The best known lower bound was established by Bollobás and Szemerédi [14], where the reader will also find a number of references on the systolic problem on graphs. Theorem 15.3. The function $\sigma(n)$ is strictly increasing, and for $n\geqslant 3$ the following lower bound holds: It follows immediately that
$$
\begin{equation}
\sigma(n) \geqslant \frac{3\ln 2}{2}\,\frac{n}{\ln n}+ o\biggl(\frac{n}{\ln n}\biggr).
\end{equation}
\tag{37}
$$
Combining (37) and (36) we obtain which coincides conceptually with the behaviour (17) of the systolic area of surface groups as depending on the genus. As we see, in the asymptotic formulae (38) and (17) the degree of the logarithm in the denominator coincides with the geometric or cohomological dimension of the groups under consideration.To the best of the author’s knowledge, the sharp asymptotic behaviour of the function $\sigma(n)$ is not known. 15.2. Combinatorial area and $\operatorname{kw}$-complexity of free groupsHere everything is quite simple: since the group $\mathbb{F}_n$ can be realised by a one-dimensional complex, the following equality holds by definition: Since all graphs with a fixed fundamental group are homotopy equivalent, it follows from the definitions in § 8 that $\operatorname{kw}(\mathbb{F}_n)=\operatorname{ct}(\Gamma)$, where $\Gamma$ is some graph of Euler characteristic $1-n$ and $\operatorname{ct}(X)$ denotes the covering type of the space $X$. The covering type of a graph as a function of its first Betti number was calculated in [37], which yields
$$
\begin{equation*}
\operatorname{kw}(\mathbb{F}_n)= \biggl\lceil \frac{3+\sqrt{1+8n}}{2}\,\biggr\rceil.
\end{equation*}
\notag
$$
15.3. The volume entropy of free groupsAs we already pointed out in § 11, Definition 11.1 is formally not suitable for free groups and must be adapted to this case. We proceed in the same way as in the beginning of § 15, where we replaced systolic area by systolic length. Consider the free group $\mathbb{F}_n$ of rank $n$. For dimensional reasons, the set of simplicial polyhedra with fundamental group $\mathbb{F}_n$ reduces to one-dimensional polyhedra, that is, graphs. If $\Gamma$ is a graph, then, applying the general Definition 10.5 (part $3^{\circ}$), where $m=1$ and the total length of the metric graph is used instead of the volume, we obtain the minimum volume entropy $\mathrm{ent}(\Gamma)$ of this graph. Since $m=1$, we obtain $\mathrm{Ent}(\Gamma)=\mathrm{ent}(\Gamma)$, and we arrive at a full analogue of Definition 11.1: Definition 15.4. The volume entropy of the group $\mathbb{F}_n$ is the quantity where $\Gamma$ ranges over all graphs with fundamental group $\mathbb{F}_n$. Since all graphs are aspherical, the entropy defined above coincides with the entropy of Bregman and Clay:
$$
\begin{equation*}
\mathrm{Ent}(\mathbb{F}_n)=\mathrm{Ent}^{(1)}(\mathbb{F}_n), \qquad n \in \mathbb{N}.
\end{equation*}
\notag
$$
Therefore, the problem of finding the volume entropy of free groups, as in the case of systolic length, is fully reduced to the problem of finding an entropy minimal graph with given fundamental group. This problem was successfully solved, and its first solution was due to Kapovich and Nagnibeda [36], who, in particular, proved the following statement. Theorem 15.5. Among all metric multigraphs whose vertices have valence at least $3$ and whose fundamental group has rank $n\geqslant 2$, any trivalent multigraph equipped with an equilateral metric has the minimum volume entropy. Here the condition that the valence is not less than $3$ is quite natural. The presence of a vertex of valence $1$ obviously increases the volume entropy of a metric graph, and vertices of valence $2$ in a metric graph can be ‘forgotten’ by a transition to a new multigraph isometric to the original one. It should be noted that Lim [44] solved the problem of the description of minimal graphs not only globally, but also in each combinatorial class, that is, in the class of graphs with a prescribed distribution of valences. Finally, in [50] the reader will find a quite different approach to the calculation of the minimum entropy of graphs. The following equality follows directly from Theorem 15.5: In conclusion, we note that in the case of free groups, statements $1^{\circ}$ of Propositions 11.4 and 14.7 are no longer true, and the inequalities from statements $2^{\circ}$ of these same propositions turn into equalities.
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\Bibitem{Bab25} |
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