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:N→N, there exists a finitely generated torsion group of intermediate growth G for which the Følner function satisfies FølG,S(n)≥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