Piecewise automatic groups

Abstract

We introduce a notion of a piecewise automatic group. Among these groups we describe a new class of groups of intermediate growth. We show that for any function $f:\mathbb{N}\to \mathbb{N}$, there exists a finitely generated torsion group of intermediate growth $G$ for which the Følner function satisfies ${\mathrm{Føl}}_{G,S}(n)\ge f(n)$ for some generating set $S$ and all sufficiently large $n$. As a corollary we see that the asymptotic entropy of simple random walks on these groups could be arbitrarily close to being linear, while the Poisson boundary is trivial