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 R\"over's simple group containing the Grigorchuk group and Thompson's group $V$ is rationally acyclic but has nontrivial Schur multiplier. We prove many more R\"over--Nekrashevych groups of self-similar groups are rationally acyclic.