Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

Bachir Bekka, Camille Francini

Published 2018-02-25Version 1

Let X be a solenoid, that is, a compact finite dimensional connected abelian group with normalized Haar measure m, and let G be a countable discrete group acting on X by continuous affine transformations. We show that the probability measure preserving action of G on (X,m) does not have the spectral gap property if and only if there exists a p(G)-invariant proper subsolenoid Y of X such that the image of G in the affine group Aff(X/Y) of X/Y is a virtually solvable group, where p(G) is the automorphism part of G. When G is finitely generated or when X is a p-adic solenoid, the subsolenoid Y can be chosen so that the image of G in Aff(X/Y) is virtually abelian. In particular, an action of a group by affine transformations on a solenoid has the spectral gap property if and only if this action is strongly ergodic.