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

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.

v3: Added Definition 2.11 and corrected a minor flaw in the original (in-line) definition. Fixed a minor error in Theorem 2.18, and improved Proposition 2.17 to compensate. 78 pages. v2: minor changes, including reorganised introduction