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