Self-similarity and limit spaces of substitution tiling semigroups

TL;DR

This paper demonstrates that the tiling semigroup of an FLC substitution tiling is self-similar and connects its limit space to the Anderson--Putnam complex, revealing a homeomorphism with the tiling's translational hull.

Contribution

It extends the concept of limit spaces from self-similar groups to self-similar semigroups in the context of substitution tilings, establishing a topological correspondence.

Findings

The tiling semigroup is self-similar.

The limit space is homeomorphic to the Anderson--Putnam complex.

The inverse limit of the limit space corresponds to the tiling's translational hull.

Abstract

We show that Kellendonk's tiling semigroup of an FLC substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson--Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.