Free subgroups in groups acting on rooted trees

Abstract

Abstract. We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary @T and a free subgroup of G with trivial stabilizer of w, or there exists w 2 @T and a free subgroup of G fixing w and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki. Mathematics Subject Classification (2010). 20E08, 20E05.

On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups

“…We call a subgroup of Aut (X * ) in which every element is a nearly finitary automorphism a nearly finitary group. Every nearly finitary automorphism is an example of a bounded automorphism (for the definition of a bounded automorphism see [21]). Nekrashevych showed ( [21], Theorem 4.4) that if the tree X * has a bounded degree of vertices, then the group of bounded automorphisms does not contain non-Abelian free subgroups.…”

“…Every nearly finitary automorphism is an example of a bounded automorphism (for the definition of a bounded automorphism see [21]). Nekrashevych showed ( [21], Theorem 4.4) that if the tree X * has a bounded degree of vertices, then the group of bounded automorphisms does not contain non-Abelian free subgroups. In the present paper, by using the characterization of groups acting faithfully on rooted trees due to Nekrashevych ([21], Theorem 3.3), we show that in the case of nearly finitary groups, there is no need for the above restriction on the degree of vertices and we obtain the following: In Sect.…”

“…Below, we recall some terminology and results due to Nekrashevych [21]. We use them to prove that nearly finitary groups do not contain non-Abelian free subgroups.…”

A finitely generated branch group of exponential growth without free subgroups

“…Sidki and Wilson constructed in [17] branch groups that contain free subgroups and hence have exponential growth. Nekrashevych proved in [13] that branch groups containing free subgroups fall into one of two cases. A paper by Brieussel [6] gives examples of groups that have a given oscillation behaviour of intermediate growth rate.…”

Growth of Schreier graphs of automaton groups

“…However, it is known that these groups do not contain non-abelian free subgroups [18,20] (but may contain free semigroups and be of exponential growth), and that groups generated by polynomial automata of degree 0 and 1 (bounded and linear automata) are amenable [4,1]. In [18] V. Nekrashevych introduced a general approach to the existence of free subgroups in automaton groups and applied it to contracting groups and to groups generated by polynomial automata. It was shown that there exists certain trichotomy for groups acting on rooted trees that involves the absence of non-abelian free subgroups as one of the three cases.…”