This article is cited in 2 scientific papers (total in 2 papers)
On algorithmic problems in Coxeter groups
V. N. Bezverkhniiab, N. B. Bezverkhnyayaab, I. V. Dobryninaab, O. V. Inchenkoab, A. E. Ustyanab
a Tula State University
b Tula State Pedagogical University
The main algorithmic problems of group theory posed by M. Dehn
are the problem of words, the problem of the conjugation of words
for finitely presented groups, and the group's isomorphism
Among the works related to the study of the M. Dehn's problems,
the most outstanding ones are the work of P. S. Novikov who proved
the undecidability of the problem of words and the conjugacy
problem for finitely presented groups as well as the
undecidability of the problem of isomorphism of groups. In this
regard, the main algorithmic problems and their various
generalizations are studied in certain classes of groups.
Coxeter groups were introduced by H. S. M. Coxeter: every
reflection group is a Coxeter group if its generating elements are
reflections with respect to hyperplanes limiting its fundamental
polyhedron. H. S. M. Coxeter listed all the reflection groups in
three-dimensional Euclidean space and proved that they are all
Coxeter groups and every finite Coxeter group is isomorphic to
some reflection group in the three-dimensional Euclidean space
which elements have a common fixed point.
In an algebraic aspect Coxeter groups are studied starting with
works by J. Tits who solved the problem of words in certain
The article describes the known results obtained in solving
algorithmic problems in Coxeter groups; the main purpose of the
paper is to analyze of the results of solving algorithmic problems
in Coxeter groups that were obtained by members of the Tula
algebraic school “Algorithmic problems of theory of the groups
and semigroups” under the supervision of V. N. Bezverkhnii.
It reviews assertions and theorems proved by the authors of the
article for the various classes of Coxeter groups: Coxeter groups
of large and extra-large types, Coxeter groups with a
tree-structure, and Coxeter groups with $n$-angled structure.
The basic approaches and methods of evidence among which the
method of diagrams worked out by van Kampen, reopened by R. Lindon
and refined by V. N. Bezverkhnii concerning the introduction of
R-cancellations, special $R$-cancellations, special ring
cancellations as well as method of graphs, method of types worked
out by V. N. Bezverkhnii, method of special set of words designed
by V. N. Bezverkhnii on the basis of the generalization of Nielsen
method for free construction of groups.
Classes of group considered in the article include all Coxeter
groups which may be represented as generalized tree structures of
Coxeter groups formed from Coxeter groups with tree structure with
replacing some vertices of the corresponding tree-graph by Coxeter
groups of large or extra-large types as well as Coxeter groups
with $n$-angled structure.
Coxeter group, algorithmic problems, diagrams.
PDF file (687 kB)
V. N. Bezverkhnii, N. B. Bezverkhnyaya, I. V. Dobrynina, O. V. Inchenko, A. E. Ustyan, “On algorithmic problems in Coxeter groups”, Chebyshevskii Sb., 17:4 (2016), 23–50
Citation in format AMSBIB
\by V.~N.~Bezverkhnii, N.~B.~Bezverkhnyaya, I.~V.~Dobrynina, O.~V.~Inchenko, A.~E.~Ustyan
\paper On algorithmic problems in Coxeter groups
\jour Chebyshevskii Sb.
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This publication is cited in the following articles:
I. V. Dobrynina, “Ob algoritmicheskikh problemakh v obobschennykh drevesnykh strukturakh grupp Kokstera”, Chebyshevskii sb., 19:2 (2018), 477–490
V. N. Bezverkhnii, I. V. Dobrynina, “O probleme obobschennoi sopryazhennosti slov v obobschennykh drevesnykh strukturakh grupp Kokstera”, Chebyshevskii sb., 19:3 (2018), 135–147
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