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 C*-algebra. We provide exact sequences to compute the homology of the groupoid and the K-theory of the C*-algebra in terms of the homology of the group and K-theory of the group C*-algebra via the transfer map and the virtual endomorphism. Complete computations are then performed for the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups and many others. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of Xin Li, we are able to show that R\"over's simple group containing the Grigorchuk group is rationally acyclic but has nontrivial Schur multiplier. We prove many more R\"over--Nekrashevych groups of self-similar groups are rationally acyclic.