Matui's AH conjecture for Graph Groupoids

Abstract

We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the K-theory of its groupoid C * -algebra.

Katsura--Exel--Pardo Groupoids and the AH~Conjecture

“…The same goes for transformation groupoids associated to odometers [Sca18], which incidentally, provided counterexamples to the other conjecture from [Mat16], namely the HK conjecture. In the recent paper [NO20], we showed that the AH conjecture holds for graph groupoids of infinite graphs, complementing Matui's result in the finite case [Mat15].…”

“…In the case that B = 0 the map ϕ : H 0 H A,0 → H 0 H A,0 coincides the inverse δ −1 of Matui's map δ from [Mat15, page 56]. See Remarks 7.6 and 8.8 in [NO20] for more on this.…”

“…As mentioned in the introduction, this paper is a follow-up to our recent paper [NO20]. We treat the same problem-namely the AH conjecture-for a related, but different, class of groupoids.…”

“…And this isomorphism is implemented by the connecting homomorphism ∂ 1 from (4.8) for B = 0. Lemma 8.6 in [NO20] (or the proof of Lemma 4.10 with B = 0) says that under this isomorphism the element…”

“…6We apologize for the conflicting notation of ϕ with the 1-cocycle from Section 3, but since the 1-cocycle makes no appearence for the rest of this section we believed it better to stick with the notation from [NO20] to make it easier to compare with results therein.…”