Periodic Points in Shifts of Finite Type Over Groups with Connections to Growth

Abstract

We develop several tools and techniques for constructing or proving the non-existence of weakly and strongly aperiodic shifts of finite type on groups. Additionally, inspired by the classification of all groups of polynomial growth and its implications for SFT-periodicity, we consider the task of surveying groups according to their possible exponential growth rates. We also propose a theory of algebraic shift spaces and pose several questions for future investigations.

Our main results are as follows: we prove that weak and strong SFT-periodicity is a commensurability invariant for all finitely generated groups, and we show that the extension of a group with a strongly aperiodic SFT by another such group has a strongly aperiodic SFT as well, provided the kernel is finitely generated. On the topic of exponential growth rates, we provide data for the growth spectrum of the free group on two generators, showing in particular that the growth spectrum is unbounded and has infinitely many limit points.