A geometric simulation theorem on direct products of finitely generated groups

Abstract

We show that every effectively closed action of a finitely generated group $G$ on a closed subset of $\{0,1{\}}^{\mathbb{N}}$ can be obtained as a topological factor of the $G$-subaction of a $(G\times H_1\times H_2)$-subshift of finite type (SFT) for any choice of infinite and finitely generated groups $H_1,H_2$. As a consequence, we obtain that every group of the form $G_1\times G_2\times G_3$ admits a non-empty strongly aperiodic SFT subject to the condition that each $G_i$ is finitely generated and has decidable word problem. As a corollary of this last result we prove the existence of non-empty strongly aperiodic SFT in a large class of branch groups, notably including the Grigorchuk group.

27 pages, 4 very beautiful figures

References in corpus (5)

• Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type
• Realization of aperiodic subshifts and uniform densities in groups
• A notion of effectiveness for subshifts on finitely generated groups
• A generalization of the simulation theorem for semidirect products
• Periodic Points on Shifts of Finite Type and Commensurability Invariants of Groups