Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

Abstract

Inspired by work of Szymik and Wahl on the homology of Higman-Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids.Applications include complete rational computations, general vanishing and acyclicity results for group homology of topological full groups as well as a proof of Matui's AH-conjecture for all minimal, ample groupoids with comparison.

Generalisations of Thompson's group V arising from purely infinite groupoids

“…This allows us to reduce the question of understanding the representations of F(G) to understanding the abelianization F(G) ab = F(G)/D(G). This abelianization can be described very concretely through Matui's AH conjecture, which has recently been verified by Xin Li [21] for the class of groupoids considered in this paper. This allows for a wealth of concrete computations: for example, we are able to describe concretely all characters of the Brin-Higman-Thompson groups nV k,r ; see Example 4.7.…”

“…2 Comparison is a notion for groupoids defined in the second paragraph of page 6 of [21]. For our purposes, it suffices to mention that comparison is automatic for purely infinite, minimal groupoids; see the comments at the top of page 5 of [21].…”

“…Since its conception, the framework of topological full groups has served as a bridge between Thompson-like groups and the groupoid models of purely infinite C*-algebras, a line of research that was pioneered by Matui [25]. In fact, promising observations by Nekrashevych [29] predate, and even motivate, the definition of topological full groups of étale groupoids by Matui. Because the underlying topological groupoids are often much more accessible to study than the Thompson-like groups themselves, this new dynamical perspective has led to progress in our understanding of the homology [21], finite presentation [20], and subgroup structure of Thompson-like groups [4].…”

“…In turn, analysing the topological full groups led us to the discovery of interesting new groups, many of which exhibit properties enjoyed by Thompson's group V . This vein of research is summarised in the table below: Very general structural properties have been studied for these examples in [27], and significant progress toward homological properties has been made in [21]. In particular, topological full groups inside this class give rise to many new examples of finitely presented, simple, infinite groups [21,20,24,26,31,33,32].…”