Homology of the Katsura-Exel-Pardo groupoid

TL;DR

This paper computes the homology of the Katsura-Exel-Pardo groupoid, demonstrating its connection to the K-theory of associated C*-algebras and confirming the (HK) conjecture, with implications for classifiable simple C*-algebras.

Contribution

It provides the first computation of the homology for this groupoid class and links it to K-theory, advancing understanding of the (HK) conjecture and groupoid C*-algebras.

Findings

Homology of the Katsura-Exel-Pardo groupoid matches the K-theory of associated C*-algebras.

Supports the (HK) conjecture for these groupoids.

Shows several classifiable simple C*-algebras are groupoid C*-algebras of this class.

Abstract

We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.