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Homology and K-theory for self-similar actions of groups and groupoids

Alistair Miller, Benjamin Steinberg

TL;DR

The paper develops a transfer-map–driven framework to compute homology of ample groupoids and K-theory of Nekrashevych C$^*$-algebras for a wide class of self-similar group and groupoid actions. Central to the method are long exact sequences and six-term exact sequences linked to virtual endomorphisms, transfer maps, and étale correspondences, extended to non-Hausdorff settings via tight groupoids. The authors apply the theory to prominent examples (Grigorchuk, Grigorchuk–Erschler, Gupta–Sidki, lamplighter, free abelian, Hanoi towers, Aleshin, etc.), proving amenability in contracting cases and deriving explicit homology and K-theory invariants, including rational acyclicity results for Röver–Nekrashevych groups. They also introduce loose faithfulness and tight kernel notions to handle quotients and Morita equivalences, enabling simplifications by passing to nicer groups. Overall, the work links group-theoretic data (transfers, virtual endomorphisms) with operator-algebraic invariants, yielding a powerful computational toolkit and broad insights into HK properties and Röver-type phenomena.

Abstract

Nekrashevych associated to each self-similar group action an ample groupoid and a $\mathrm{C}^\ast$-algebra. We perform complete computations of the homology of the groupoid and the K-theory of the $\mathrm{C}^\ast$-algebra for a myriad of examples, including the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups, and self-similar actions of free abelian groups and lamplighter groups. The key development is the construction, for arbitrary self-similar group actions, of long exact sequences which compute the homology and K-theory in terms of the homology of the group and K-theory of the group $\mathrm{C}^\ast$-algebra via the transfer map and the virtual endomorphism. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of X.~Li, we are able to show that Röver's simple group containing the Grigorchuk group and Thompson's group $V$ is rationally acyclic but has nontrivial Schur multiplier. We prove many more Röver--Nekrashevych groups of self-similar groups are rationally acyclic.

Homology and K-theory for self-similar actions of groups and groupoids

TL;DR

The paper develops a transfer-map–driven framework to compute homology of ample groupoids and K-theory of Nekrashevych C-algebras for a wide class of self-similar group and groupoid actions. Central to the method are long exact sequences and six-term exact sequences linked to virtual endomorphisms, transfer maps, and étale correspondences, extended to non-Hausdorff settings via tight groupoids. The authors apply the theory to prominent examples (Grigorchuk, Grigorchuk–Erschler, Gupta–Sidki, lamplighter, free abelian, Hanoi towers, Aleshin, etc.), proving amenability in contracting cases and deriving explicit homology and K-theory invariants, including rational acyclicity results for Röver–Nekrashevych groups. They also introduce loose faithfulness and tight kernel notions to handle quotients and Morita equivalences, enabling simplifications by passing to nicer groups. Overall, the work links group-theoretic data (transfers, virtual endomorphisms) with operator-algebraic invariants, yielding a powerful computational toolkit and broad insights into HK properties and Röver-type phenomena.

Abstract

Nekrashevych associated to each self-similar group action an ample groupoid and a -algebra. We perform complete computations of the homology of the groupoid and the K-theory of the -algebra for a myriad of examples, including the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups, and self-similar actions of free abelian groups and lamplighter groups. The key development is the construction, for arbitrary self-similar group actions, of long exact sequences which compute the homology and K-theory in terms of the homology of the group and K-theory of the group -algebra via the transfer map and the virtual endomorphism. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of X.~Li, we are able to show that Röver's simple group containing the Grigorchuk group and Thompson's group is rationally acyclic but has nontrivial Schur multiplier. We prove many more Röver--Nekrashevych groups of self-similar groups are rationally acyclic.
Paper Structure (32 sections, 67 theorems, 90 equations, 2 figures)

This paper contains 32 sections, 67 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $\mathrm{C}^\ast$-algebras and groupoids associated to self-similar groupoids
  3. Self-similar groupoids
  4. $\mathrm{C}^\ast$-algebras, inverse semigroups and ample groupoids
  5. Faithful quotients of self-similar groupoids
  6. Amenability
  7. The Dramatis Personæ
  8. Miscellaneous examples
  9. Multispinal self-similar groups
  10. Solvable self-similar groups
  11. Tools for computing homology and K-theory
  12. Homology of ample groupoids
  13. Six-term and long exact sequences
  14. Étale correspondences
  15. The transfer map
  16. ...and 17 more sections

Key Result

Theorem 1

Let $(G,X,\sigma)$ be a self-similar group action over a finite alphabet $X$. For $x\in X$, let $\sigma_x\colon G_x\to G$ be the virtual endomorphism $g\mapsto g|_x =\sigma(g,x)$. Then there is a long exact sequence where $\Phi_n = \sum_{x \in T} H_n(\sigma_x)\circ \mathrm{tr}^{G}_{G_x}$ for any transversal $T$ to $G\backslash X$.

Figures (2)

  • Figure 1: A commutative diagram with exact rows and columns
  • Figure 2: A commutative diagram with exact rows and columns

Theorems & Definitions (151)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem : Li, Corollaries C and D li2022ample
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Definition 2.1: Étale correspondence
  • Definition 2.2: Self-similar groupoid action via correpondences
  • ...and 141 more