On the spectral theory of groups of automorphisms of $S$-adic nilmanifolds

TL;DR

This paper characterizes when groups of automorphisms of $S$-adic nilmanifolds have spectral gaps in their Koopman representations, linking spectral properties to invariant quotient solenoids and group actions.

Contribution

It provides a precise criterion for the absence of spectral gaps in automorphism groups of $S$-adic nilmanifolds, connecting spectral theory with group actions on quotient solenoids.

Findings

Spectral gap exists iff no non-trivial invariant quotient solenoid with virtually abelian action.

Spectral gap characterized by the absence of certain invariant quotient structures.

Provides a complete classification of automorphism groups based on spectral properties.

Abstract

Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique translation invariant probability measure $\mu.$ We characterize the countable groups $\Gamma$ of automorphisms of $X$ for which the Koopman representation $\kappa$ on $L^2(X,\mu)$ has a spectral gap. More specifically, we show that $\kappa$ does not have a spectral gap if and only if there exists a non-trivial $\Gamma$-invariant quotient solenoid (that is, a finite-dimensional, connected, compact abelian group) on which $\Gamma$ acts as a virtually abelian group.