Rigidity, Generators and Homology of Interval Exchange Groups

Abstract

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