
Brief communications
An ergodic theorem for actions of Fuchsian groups
A. I. Bufetov^{abc}, A. V. Klimenko^{ad}, C. Series^{e} ^{a} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^{b} Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
^{c} CNRS, Institut de Mathématiques de Marseille,
Marseille, France
^{d} HSE University
^{e} University of Warwick, Coventry, UK
Received: 03.02.2023
The main result of this note, Theorem 1, establishes the pointwise convergence of spherical averages for actions of Fuchsian groups.
Let $G$ be a finitely generated Fuchsian group. Let $\mathcal R$ be its fundamental domain, possibly with vertices or edges on the boundary of the hyperbolic disc $ \mathbb D$, and let ${\mathbf{T}_{\mathcal{R}}}=\{g{\mathcal R}\colon g\in G\}$ be the corresponding tessellation of $\mathbb D$. We say that ${\mathcal R}$ has even corners if the geodesic extension of every side of ${\mathcal R}$ if entirely contained in the union of the boundaries of all domains $g{\mathcal R}\in{\mathbf{T}_{\mathcal{R}}}$.
Let $G_0$ be the symmetric set of generators mapping $\mathcal R$ to adjacent domains in ${\mathbf{T}_{\mathcal{R}}}$. For $g\in G$ denote by $g$ the length of the shortest word in $G_0$ representing $g$. Let $S(n)$ be the sphere of radius $n$ in $G$: $S(n)=\{g\in G\colon g=n\}$.
Suppose that $G$ acts on a probability space $(X, \mu)$ by measurepreserving transformations $T_g$, $g\in G$. Given a function $f\in L^1(X,\mu)$, consider the spherical averages
$$
\begin{equation*}
\mathbf{S}_n(f)=\frac1{\#S(n)} \sum_{g\in S(n)} f\circ T_g.
\end{equation*}
\notag
$$
Let $v\in{\mathbb D}$ be a vertex of $\mathbf{T}_{\mathcal{R}}$. If ${\mathcal R}$ has even corners, then in a small neighbourhood of $v$ the boundary of ${\mathbf{T}_{\mathcal{R}}}$ consists of $n = n(v)$ geodesic segments intersecting at $v$ and dividing our neighbourhood into $2n(v)$ sectors. Let $N(\mathcal R)$ denote the number of sides of $\mathcal R$ inside $\mathbb D$. If $G_0$ has an elliptic element of order two, then we consider its fixed point as a vertex of $\mathcal R$. We need the following assumption on ${\mathcal R}$.
Assumption 1. The domain ${\mathcal R}$ has even corners. Furthermore, either $N({\mathcal R})\geqslant 5$, or ${\mathcal R}$ is noncompact and $N({\mathcal R})\in\{3,4\}$, or ${\mathcal R}$ is compact, $N({\mathcal R})=4$, and ${\mathcal R}$ does not have two opposite vertices $v$ and $v'$ such that $n(v)=n(v')=2$.
Set $L\log L(X,\mu)=\biggl\{f \in L^1\colon \displaystyle\int f \log ^+f\,d\mu < \infty\biggr\}$.
Theorem 1. Let $G$ be a nonelementary Fuchsian group with fundamental domain $\mathcal R$ satisfying Assumption 1. Let $G_0$ be the symmetric set of generators of $G$ defined above. Let $G$ act on a Lebesgue probability space $(X,\mu)$ by measurepreserving transformations. Denote by $\mathcal I_{G_0^2}$ the $\sigma$algebra of sets invariant under all maps $T_{g_1g_2}$, $g_1,g_2\in G_0$. Then for any function $f\in L\log L(X,\mu)$, as $n\to\infty$,
$$
\begin{equation*}
\mathbf{S}_{2n}(f)\to\mathsf E(f\mid \mathcal I_{G_0^2})\quad \textit{almost surely and in } L^1,
\end{equation*}
\notag
$$
where $\mathsf E(f\mid\mathcal I_{G_0^2})$ is the expectation of $f$ relative to the $\sigma$algebra $\mathcal I_{G_0^2}$.
Our proof extends the argument from [3], where the convergence of the spherical averages was established for actions of free groups. The main step in the proof of Theorem 1 is the construction of a new Markov coding for a Fuchsian group satisfying Assumption 1.
The first results on the convergence of spherical averages for Gromov hyperbolic groups, which were obtained under strong exponential mixing assumptions on the action, were due to Fujiwara and Nevo [5]. A convenient method proposed by Grigorchuk [6], Thouvenot (oral communication), and in [2] for proving ergodic theorems for actions of free semigroups and groups is to associate with the group action a Markov operator $P$ on a suitable function space. The convergence of spherical averages is thus related to the convergence of the powers $P^n f$ of this Markov operator. The proof of the convergence for free groups in [3] used Rota’s ‘Alternierende Verfahren’ [7], that is, the convergence of $(P^*)^n P^n f$. To deduce the convergence of $P^{2n}f$ one needs a relation between $P$ and $P^*$: see [3], Proposition 3. The source for this relation is the following symmetry condition for the underlying Markov coding for $G$.
For the general Fuchsian group the earlier coding by Bowen and Series [1], [4] does not satisfy this symmetry condition. Recently Wroten [8] introduced a new approach, which is based on the simultaneous consideration of all shortest words representing a given element $g$. The main step in our proof of Theorem 1 is to show that for Fuchsian groups collections of all shortest paths are generated by a symmetric Markov coding such that the generating Markov chain is irreducible and has a trivial symmetric $\sigma$algebra.



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Citation:
A. I. Bufetov, A. V. Klimenko, C. Series, “An ergodic theorem for actions of Fuchsian groups”, Uspekhi Mat. Nauk, 78:3(471) (2023), 179–180; Russian Math. Surveys, 78:3 (2023), 566–568
Linking options:
https://www.mathnet.ru/eng/rm10099https://doi.org/10.4213/rm10099e https://www.mathnet.ru/eng/rm/v78/i3/p179

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