Dmytro Savchuk

Some Graphs Related to Thompson’s Group F

The Schreier graphs of Thompson’s group F with respect to the stabilizer of 1/2 and generators x 0 and x 1 , and of its unitary representation in L 2 ([0, 1]) induced by the standard action on the interval [0, 1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x n v, where n ≥ 0 and v is any word over the alphabet {x 0 , x 1 }, is constructed. It is proved that the latter graph is non-amenable.

A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, no such examples have been constructed in some classes, as groups generated by non reversible automata. It was recently shown that 2-state reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state reversible Mealy automata, using new original techniques. The results rely on a fine analysis of associated orbit trees and a new characterization of the existence of elements of infinite order.

Affine automorphisms of rooted trees

We introduce a class of automorphisms of rooted d-regular trees arising from affine actions on their boundaries viewed as infinite dimensional modules

Ergodic decomposition of group actions on rooted trees

{\mathbb {Z}}_d^{\infty }

ON RESTRICTING SUBSETS OF BASES IN RELATIVELY FREE GROUPS

Zd∞. This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from this class and show that it is isomorphic to the index 2 extension of the rank 2 lamplighter group

Classification of groups generated by 3-state automata over a 2-letter alphabet

{\mathbb {Z}}_2^2\wr {\mathbb {Z}}