Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

Abstract

In this paper, we will study representations of the continuous full group Γ A of a one-sided topological Markov shift (X A , σ A ) for an irreducible matrix A with entries in {0, 1} as a generalization of Higman-Thompson groups V N , 1 < N ∈ N. We will show that the group Γ A can be represented as a group Γ

Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

“…We will next show the identity d ξ h (τ ) • ξ h (τ −1 ) = −d ξ h (τ −1 ) . Since ξ h (τ ) −1 = ξ h (τ −1 ), we have by using [20,Lemma 7.7],…”

“…For each τ ∈ Γ A , the function d τ : X A −→ Z is defined by d τ = l τ − k τ . It does not depend on the choice of l τ and k τ ( [20,Lemma 7.6]). The group Γ A is a countably infinite discrete non-amenable group ( [11], see [21] for more general results).…”

“…He also found simplicity condition on th groups Γ A and pointed out an interesting relationship to Thompson groups and Higman-Thompson groups (see [22], cf. [20]). For the Cuntz-Krieger algebra O A with its canonical generating partial isometries S 1 , .…”

Higman-Thompson groups from self-similar groupoid actions

“…In particular, rα, g, β; βµs ´1 " rβ, g ´1, α; αpg ¨µqs. The topology on GpG, Eq is generated by the compact open bisections of the form Zpα, g, β; Uq " trα, g, β; ξs P GpG, Eq : ξ P Uu, where U is an open compact subset of Zpβq " βE 8 . Since any open set U Ď Zpβq is a disjoint union U " ğ i Zpβγ i q where γ i P E ˚with rpγ i q " spβq, we get Zpα, g, β; Uq " ğ i Zpα, g, β; Zpβγ i qq " ğ i Zpαpg ¨γi q, g| γ i , βγ i ; Zpβγ i qq.…”

“…A split of a The set of all homeomorphisms of E 8 defined by such tables is a subgroup of HomeopE 8 q, called the Higman-Thompson group of E and denoted by V E . In [8], the authors define tables associated to a onesided topological Markov shift pX A , σ A q for an irreducible N ˆN matrix A with entries in t0, 1u. They studied the corresponding Higman-Thompson group and proved that it is isomorphic to the topological full group of the groupoid associated to pX A , σ A q.…”