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

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.