Periodic points and residual finiteness of automorphism groups of subshifts

Abstract

If totally periodic points are dense in a subshift $X$, its automorphism group is residually finite. We show a weak converse: if periodic points are not dense in a subshift $X$, then the automorphism group of $X\times Y$ is not residually finite for full shifts $Y$ (and sufficiently full-shift-like subshifts). On the other hand, we show that the automorphism group of a block gluing ${\mathbb{Z}}^2$-subshift is always locally embeddable in finite groups (thus sofic). Hochman recently constructed a strongly irreducible ${\mathbb{Z}}^2$-subshift with no periodic points. Combining our result with this example gives a strongly irreducible ${\mathbb{Z}}^2$-subshift whose automorphism group is not residually finite, which solves a question of Coornaert and Ceccherini-Silberstein.

27 pages, 5 figures