Strong relative property $(T)$ and spectral gap of random walks

TL;DR

This paper investigates strong relative property (T) for group actions on Lie groups and solenoids, establishing criteria for spectral gaps and analyzing the impact of group structure and automorphisms.

Contribution

It introduces new conditions under which strong relative property (T) holds for pairs involving automorphism groups of Lie groups and solenoids, extending previous results.

Findings

Strong relative property (T) holds for certain automorphism pairs with no compact factors.

Characterization of when pairs involving Lie groups and their radicals have strong relative property (T).

Applications to spectral gap properties of unitary representations.

Abstract

We consider strong relative property $(T)$ for pairs $(\Ga, G)$ where $\Ga$ acts on $G$. If $N$ is a connected Lie group and $\Ga$ is a group of automorphisms of $N$, we choose a finite index subgroup $\Ga ^0$ of $\Ga$ and obtain that $(\Ga, [\Ga ^0, N])$ has strong relative property $(T)$ provided Zariski-closure of $\Ga$ has no compact factor of positive dimension. We apply this to obtain the following: $G$ is a connected Lie group with solvable radical $R$ and a semisimple Levi subgroup $S$. If $S_{nc}$ denotes the product of noncompact simple factors of $S$ and $S_T$ denotes the product of simple factors in $S_{nc}$ that have property $(T)$, then we show that $(\Ga, R)$ has strong relative property $(T)$ for a Zariski-dense closed subgroup of $S_{nc}$ if and only if $R=[S_{nc},R]$. The case when $N$ is a vector group is discussed separately and some interesting results are proved. We also considered actions on solenoids $K$ and proved that if $\Ga$ acts on a solenoid $K$, then $(\Ga, K)$ has strong relative property $(T)$ under certain conditions on $\Ga$. For actions on solenoids we provided some alternatives in terms of amenability and strong relative property $(T)$. We also provide some applications to the spectral gap of $\pi (\mu)=\int \pi (g) d\mu (g)$ where $\pi$ is a certain unitary representation and $\mu$ is a probability measure.