Faithful group actions and Schreier graphs

Array

Authors

  • M. Fedorova Taras Shevchenko National University, 64/13 Volodymyrska str., 01601, Kyiv, Ukraine

DOI:

https://doi.org/10.15330/cmp.9.2.202-207

Keywords:

group action, faithful action, Schreier graph, free product, automaton permutation

Abstract

Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical properties of corresponding group actions. In particilar, they are widely used in order to check amenability of different classed of groups. In the present paper Schreier graphs are utilized to construct new examples of faithful actions of free products of groups. Using Schreier graphs of group actions a sufficient condition for a group action to be faithful is presented. This result is applied to finite automaton actions on spaces of words i.e. actions defined by finite automata over finite alphabets. It is shown how to construct new faithful automaton presentations of groups upon given such a presentation. As an example a new countable series of faithful finite automaton presentations of free products of finite groups is constructed. The obtained results can be regarded as another way to construct new faithful actions of  groups  as soon as at least one such an action is provided.

Downloads

Additional Files

Published

2018-01-03

How to Cite

(1)
Fedorova, M. Faithful Group Actions and Schreier Graphs: Array. Carpathian Math. Publ. 2018, 9, 202-207.

Issue

Section

Scientific articles