The large scale geometry of strongly aperiodic subshifts of finite type

TL;DR

This paper explores the large-scale geometry of strongly aperiodic subshifts of finite type on groups, establishing invariance properties under quasi-isometry and linking them to group properties like ends and rigidity.

Contribution

It demonstrates that strongly aperiodic SFTs exist on groups with at least two ends and that their existence and the domino problem's decidability are quasi-isometry invariants for certain groups.

Findings

Groups with ≥2 ends admit strongly aperiodic SFTs.

Existence of strongly aperiodic SFTs is a quasi-isometry invariant.

Decidability of the domino problem is a quasi-isometry invariant.

Abstract

A subshift on a group G is a closed, G-invariant subset of A^G, for some finite set A. It is said to be a subshift of finite type (SFT) if it is defined by a finite collection of 'forbidden patterns', to be strongly aperiodic if all point stabilizers are trivial, and weakly aperiodic if all point stabilizers are infinite index in G. We show that groups with at least 2 ends have a strongly aperiodic SFT, and that having such an SFT is a QI invariant for finitely presented torsion free groups. We show that a finitely presented torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino problem on G asks whether the SFT specified by a given set of forbidden patterns is empty. We show that decidability of the domino problem is a QI invariant.