Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them

TL;DR

This paper explores the existence and significance of strongly aperiodic subshifts of finite type (SFTs) on hyperbolic groups, explaining their properties, construction, and the underlying ideas behind recent proofs.

Contribution

It provides an accessible exposition of the conditions under which hyperbolic groups admit strongly aperiodic SFTs and discusses the ideas involved in the proof of this characterization.

Findings

Hyperbolic groups with at most one end admit strongly aperiodic SFTs.

The paper clarifies the concept and potential applications of strongly aperiodic SFTs.

It explains the proof techniques used in establishing the existence of such SFTs.

Abstract

D. B. Cohen, C. Goodman-Strauss, and the author proved that a hyperbolic group admits an "SA SFT" if and only if it has at most one end. This paper has two distinct parts: the first is a conversation explaining what an SA SFT is and how they may be of use. In the second part I attempt to explain both old and new ideas that go into the proof. References to specific claims in the original paper are given, with the hope that any interested reader may be able to find the details there more accessible after reading this exposition.