Self-similarity and limit spaces of substitution tiling semigroups
James Walton
University of Nottingham, Nottingham, UKMichael F. Whittaker
University of Glasgow, Glasgow, UK
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Abstract
We show that Kellendonk’s tiling semigroup of a finite local complexity 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.
Cite this article
James Walton, Michael F. Whittaker, Self-similarity and limit spaces of substitution tiling semigroups. Groups Geom. Dyn. (2024), published online first
DOI 10.4171/GGD/807