# A geometric simulation theorem on direct products of finitely generated groups

## TL;DR

This paper demonstrates that effectively closed actions of finitely generated groups can be represented as factors of subshifts of finite type on product groups, leading to the existence of strongly aperiodic SFTs in various complex groups.

## Contribution

It introduces a geometric simulation theorem for direct products of finitely generated groups, enabling the construction of strongly aperiodic SFTs in these groups.

## Key findings

- Every effectively closed action of a finitely generated group is a factor of a subshift of finite type.
- Any product of three finitely generated groups with decidable word problems admits a strongly aperiodic SFT.
- Existence of strongly aperiodic SFTs in large classes of branch groups, including the Grigorchuk group.

## 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.