Rigidity, Generators and Homology of Interval Exchange Groups

Abstract

Let $\Gamma$ be a dense countable subgroup of $\mathbb{R}$. Then, consider $IE(\Gamma)$; the group of piecewise linear bijections of $[0,1]$ with finitely many angles, all in $\Gamma$. We introduce and systematically study a family of partial transformation groupoids coming from inverse semigroups, $\mathcal{G}_\Gamma$, that realise $IE(\Gamma)$ as a topological full group. This new perspective on the groupoid models $\mathcal{G}_\Gamma$ of $IE(\Gamma)$ allows us to better understand the underlying C*-algebras and to compute homology. We show that $H_*(\mathcal{G}_\Gamma)=H_{*+1}(\Gamma)$. We show $C^*_r(\mathcal{G}_\Gamma)$ is classifiable in the sense of the Elliott classification program of C$^*$-algebras. We then classify these groups via the Elliott invariant, showing $IE(\Gamma) \cong IE(\Gamma') \Leftrightarrow \Gamma=\Gamma'$ as subsets of $\mathbb{R}$. We relate the K-Theory of the reduced C$^*$ -algebras to groupoid homology via Matui's HK Conjecture. We relate the homology of $IE(\Gamma)$ to the homology of $\Gamma$ using the recent framework developed by Li. We investigate in greater detail three key cases, namely if $\Gamma \subset \mathbb{Q}$, if $\Gamma \cong \mathbb{Z}^n$, and if $\Gamma$ is a ring. For these three cases, we study homology in greater detail and find explicit generating sets.