Jérémie Brieussel

Behaviors of entropy on finitely generated groups

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any 12≤α≤β≤1, there is a group Γ with measure μ equidistributed on a finite generating set such that lim inflogHΓ,μ(n)logn=α,lim suplogHΓ,μ(n)logn=β. The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk Yn on such groups are also evaluated, providing an example of group with return probability satisfying lim inflog|logP(Yn=Γ1)|logn=13,lim suplog|logP(Yn=Γ1)|logn=1 and drift satisfying lim inflogE∥Yn∥logn=12,lim suplogE∥Yn∥logn=1.

An automata group of intermediate growth and exponential activity

Abstract We give a new example of an automata group of intermediate growth. It is generated by an automaton with four states on an alphabet with eight letters. This automata group has exponential activity and its limit space is not simply connected.