Strongly aperiodic subshifts of finite type on hyperbolic groups

Abstract

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

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

“…Much of this paper reviews old and well-known results. In order to facilitate the reading of [9], we included many references to that work. This is not an indication that all the claims are original.…”

“…Some show existence of an SA SFT, and some show that the Domino Problem is undecidable (implying the existence of a weakly aperiodic set of tiles, but not implying the existence of an SA SFT). More results can be found in [9]. The 2018 survey [1] contains many results about the Domino Problem.…”

“…If g is a torsion element then C = 0. Conversly, if g has infinite order then C = 0 (Lemma 9.1 of [9]). This simple observation will prove quite useful.…”

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In [9] 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.

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“…It has two parts: in the Part 1 we discuss the reasons for constructing SA SFTs and explain what they are. In Part 2 we explain the basic ideas behind the construction which led to the theorem of D. B. Cohen, C. Goodman-Strauss, and the author [9]:…”