Left regular representations of Garside categories I. C^*-algebras and groupoids

Abstract

We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.

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

“…In contrast to the above corollary, in many cases F(G) does have a solvable word problem, for example the topological full groups from many groupoids that arise as left regular representations of Garside Categories (see [20,Corollary C]) are known to have solvable word problem. Therefore, we obtain infinitely many nonisomorphic groups spanning many families that demonstrate this boundary between the word problem and the generalised word problem.…”

“…(3) Groupoids arising from Beta expansions [24]. (4) Certain groupoids arising from left regular representations of Garside categories [20]; see also [22, Theorem A].…”

“…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].…”

“…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].…”

“…Also, it allows us to conclude that the generalised word problem is not solvable for F(G). Crucially, this means that many of these topological full groups, for example everything in the Garside category framework introduced by Li in [20] (a broad class including all Brin-Higman-Thompson groups and Stein groups), have solvable word problem but not solvable generalised word problem. For the precise definition of the (generalised) word problem, see Definition 3.9.…”