Eventually self-similar groups acting on fractals built from almost expanding hyperedge replacement systems inherit finiteness properties from their self-similar building blocks, yielding F∞ for airplane and dendrite rearrangement groups.
Higman-Thompson groups from self-similar groupoid actions
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abstract
Given a self-similar groupoid action $(G,E)$ on a finite directed graph, we prove some properties of the corresponding ample groupoid of germs $\mathcal G(G,E)$. We study the analogue of the Higman-Thompson group associated to $(G,E)$ using $G$-tables and relate it to the topological full group of $\mathcal G(G,E)$, which is isomorphic to a subgroup of unitaries in the algebra $C^*(G,E)$. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for $\mathcal G(G,E)$ in some particular cases.
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Eventually Self-Similar Groups acting on Fractals
Eventually self-similar groups acting on fractals built from almost expanding hyperedge replacement systems inherit finiteness properties from their self-similar building blocks, yielding F∞ for airplane and dendrite rearrangement groups.