Numerical upper bounds on growth of automaton groups

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

Cascade Products of Stochastic Automata

Stochastic Moore automata have in opposition to stochastic Mealy automata the same capabilities as general stochastic automata, but have the advantage that they are easier to access than their pure stochastic counterparts. Cascade decomposition of automata leads to a loop-free partitioning and in this way contributes to the analysis of automata. This paper shows that stochastic Moore automata can be decomposed into cascade products of stochastic Moore automata under mild conditions

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, all such examples have been constructed as groups generated by non-reversible automata. Moreover, 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.