A general framework for quasi-isometries in symbolic dynamics beyond groups

Abstract

We introduce an algebraic structure which encodes a collection of countable graphs through a set of states, generators and relations. These structures, which we call blueprints, can capture standard algebraic objects such as groups, monoids or small categories, as well as geometric tiling spaces with finite local complexity. We provide a general framework for symbolic dynamics on blueprints under a partial monoid action, and for transferring invariants of their symbolic dynamics through quasi-isometries. In particular, we show that the undecidability of the domino problem, the existence of strongly aperiodic subshifts of finite type, and the existence of subshifts of finite type without computable points are all quasi-isometry invariants for finitely presented blueprints. As an application of this model, we show that two variants of the domino problem for geometric tilings of $\mathbb{R}^d$ are undecidable for $d \geq 2$ on any underlying tiling space with finite local complexity.