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

TL;DR

This paper characterizes when group actions on solenoids have the spectral gap property, linking it to the existence of invariant subsolenoids and the algebraic structure of the group, with implications for strong ergodicity.

Contribution

It provides a complete characterization of spectral gap property for affine group actions on solenoids, connecting it to the group's algebraic properties and invariant substructures.

Findings

Spectral gap property holds iff no invariant proper subsolenoid with certain algebraic conditions exists.

For finitely generated groups or p-adic solenoids, the subsolenoid can be chosen so the group action is virtually abelian.

Spectral gap property is equivalent to strong ergodicity for these actions.

Abstract

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.