Homology and topological full groups of étale groupoids on totally disconnected spaces

By H. Matui. 2009

193 citations recorded by the source corpus.

For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H0 if and only if there exists an element in the topological full group that maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H1 to K1 of the groupoid C*‐algebra. Explicit computations of homology groups of AF groupoids and étale groupoids arising from subshifts of finite type are also given.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e12d42c664db792c92eed925cc9b6d28219b2f0e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, background): “Matui [21] defines Kakutani equivalence for ample groupoids G and H with compact unit spaces: G and H are Kakutani equivalent if there are full clopen subsets X ⊆ G and Y ⊆ H such that G|X ∼= H|Y .”
Palindromic subshifts and simple periodic groups of intermediate growth (2016, background): “ of all homeomorphisms h : X −→ X such that for every ζ ∈ X there exists a neighborhood U of ζ and an element g ∈ G such that h|U = g|U. Topological full groups were introduced in [4]. See the papers [10, 12, 13, 11] for various properties of topological full groups of group actions and ´etale group”
Cartan subalgebras in C*-algebras. Existence and uniqueness (2017, unknown): “(QI1) Γ and Λ are quasi-isometric; (QI2) Γ y βΓ and Λ y βΛ are Kakutani equivalent, in the sense of [33, 32]; (QI3) Γ y βΓ and Λ y βΛ are stably continuously orbit equivalent, in the sense of [7, 32]; (QI4) There exist projections p ∈ C(βΓ) ⊆ C(βΓ)⋊r Γ and q ∈ C(βΛ) ⊆ C(βΛ)⋊r Λ which are full in C(β”
Almost Finiteness and the Small Boundary Property (2018, unknown): “When X is zero-dimensional this agrees with Matui’s original notion of almost finiteness for second countable étale groupoids with compact zero-dimensional unit spaces [34].”
Topological full groups of one-sided shifts of finite type (2012, background): “The notion of topological full groups was later generalized to the setting of essentially principal étale groupoids G on Cantor sets in [31].”
Etale groupoids arising from products of shifts of finite type (2015, background): “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].”
Dimension, comparison, and almost finiteness (2017, background): “Matui showed in [27] that almost finiteness for a second countable étale groupoid G with compact zero-dimensional unit space has several implications for the homology groups H(G) and their relation to both the topological full group JGK and the K-theory of the reduced groupoid C-algebra of G.”
Reconstruction of groupoids and C*-rigidity of dynamical systems (2017, background): “If U1 and U2 are compact open, G1 and G2 are Kakutani equivalent as defined in [31].”
Partial transformation groupoids attached to graphs and semigroups (2016, background): “Apart from purely infinite groupoids, Matui also introduced almost finite ones in [26].”
Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras (2007, background): “..,1) ∈ ZN in BF(At) by uA. By [4, Proposition 3.1], K0(OA) is isomorphic to BF(At) and the class of the unit of OA maps to uA under this isomorphism. And K1(OA) is isomorphic to Ker(id−At) on ZN. In [15], it has been shown that these groups naturally arise from the homology theory of ´etale groupoi”
Simple groups of dynamical origin (2015, background): “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”
Ample groupoids: equivalence, homology, and Matui's HK conjecture (2018, background): “It is not stated in [44] or in [32] that similarity of groupoids is an equivalence relation, but this is standard (it is essentially the argument that category equivalence is an equivalence relation).”