Abstract
It is proven that Matui's AH~conjecture is true for Katsura--Exel--Pardo groupoids $\mathcal{G}_{A,B}$ associated to integral matrices $A$ and $B$. This conjecture relates the topological full group of an ample groupoid with the homology groups of the groupoid. We also give a criterion under which the topological full group $[[\mathcal{G}_{A,B}]]$ is finitely generated.
Highlights
The AH conjecture is one of two conjectures formulated by Matui in [8] concerning certain ample groupoids over Cantor spaces
This conjecture predicts that the abelianization of the topological full group of such a groupoid together with its first two homology groups fit together in an exact sequence as follows: H0(G) ⊗ Z2 −−−j−→ G ab −−−Ia−b−→ H1(G) −−−→ 0
The same goes for transformation groupoids associated to odometers [14], which incidentally provided counterexamples to the other conjecture from [8], namely, the HK conjecture
Summary
The AH conjecture is one of two conjectures formulated by Matui in [8] concerning certain ample groupoids over Cantor spaces. In the recent paper [12], we showed that the AH conjecture holds for graph groupoids of infinite graphs, complementing Matui’s result in the finite case [7]. The construction of Exel and Pardo encompassed the Katsura algebras They realized that the matrices A and B could be used to describe a self-similar action by the integer group Z on the graph whose adjacency matrix is A in such a way that the associated C∗-algebra. We emphasize that the Katsura–Exel–Pardo groupoids are merely prominent special cases of the tight groupoids constructed from self-similar graphs in [1] This construction was further generalized to non-row-finite graphs in [2].
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