Topological full groups of one-sided shifts of finite type

By H. Matui. 2012

161 citations recorded by the source corpus.

We explore the topological full group [[G]] of an essentially principal etale groupoid G on a Cantor set. When G is minimal, we show that [[G]] (and its certain normal subgroup) is a complete invariant for the isomorphism class of the etale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D([[G]]) is shown to be simple. The etale groupoid G arising from a one-sided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such G, [[G]] is thought of as a generalization of the Higman-Thompson group. We prove that [[G]] is of type F_\infty, and so in particular it is finitely presented. This gives us a new infinite family of finitely presented infinite simple groups. Also, the abelianization of [[G]] is calculated and described in terms of the homology groups of G.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b9e839cb9714511be860f90e47cf2d3ca54aca04

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Etale groupoids arising from products of shifts of finite type (2015, methodology): “In the case that the unit space G(0) is a Cantor set, we investigated connections between the homology groups and dynamical properties of G in [27, 28].”
Ample groupoids: equivalence, homology, and Matui's HK conjecture (2018, background): “In a series of recent papers [32, 33, 34], Matui has advanced a conjecture that if G is a minimal effective ample Hausdorff groupoid with compact unit space then K0(C ∗ r (G)) is isomorphic to the direct sum of the even homology groups H2n(G) of the groupoid as defined by Crainic and Moerdijk [11], ”
Partial transformation groupoids attached to graphs and semigroups (2016, unknown): “Furthermore, we study which partial transformation groupoids of graphs and semigroups are purely infinite, in the sense of [27].”
Simple groups of dynamical origin (2015, methodology): “There are many examples of topological full groups in this sense: the Thompson groups [4], Monod’s examples [23], a version of Lodha and Moore’s example [17], groups associated with expanding maps [25], diagram groups [9], etc. Topological full groups of étale groupoids were studied by Matui in a s”
Finite Factor Representations of Higman-Thompson groups (2012, methodology): “In Section 3, we apply our general results to the Higman-Thompson groups and to the full groups of irreducible shifts of finite type [9].”