Automorphism groups of random substitution subshifts (Q6634393)

Random substitutions are a relatively new object of study in symbolic dynamics, generalizing the notion of (deterministic) substitutions). They are of particular interest in the study of mathematical quasicrystals (see [\textit{C. Godrèche} and \textit{J. M. Luck}, J. Stat. Phys. 55, No. 1--2, 1--28 (1989; Zbl 0717.05025)]). For an introduction to random substitutions in the context of symbolic dynamics, see the paper of \textit{D. Rust} and \textit{T. Spindeler} [Indag. Math., New Ser. 29, No. 4, 1131--1155 (2018; Zbl 1409.37023)].\N\NIn the paper under review the authors prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Thus, such groups are countable, non-amenable and non-residually finite.\N\NTo show this, the authors introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, they need a more refined notion of recognisable words in order to understand their automorphisms. They show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.