Inducing Super-Approximation

TL;DR

This paper proves that under certain algebraic and spectral gap conditions, the property of spectral gap extends from a subgroup to a larger group in the context of linear groups over integers and p-adic integers.

Contribution

It establishes a new induction principle for spectral gaps in linear groups, linking subgroup properties to larger groups via algebraic and topological conditions.

Findings

Spectral gap property extends from subgroup to larger group under specified conditions.

Algebraic normal subgroup condition is crucial for the induction of spectral gap.

Results apply to finitely generated subgroups of linear groups over integers and p-adic integers.

Abstract

Let $\Gamma_2\subseteq \Gamma_1$ be finitely generated subgroups of ${\rm GL}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let $\mathbb{G}_i$ be the Zariski-closure of $\Gamma_i$ in $({\rm GL}_{n_0})_{\mathbb{Q}}$, $\mathbb{G}_i^{\circ}$ be the Zariski-connected component of $\mathbb{G}_i$, and let $G_i$ be the closure of $\Gamma_i$ in $\prod_{p\nmid q_0}{\rm GL}_{n_0}(\mathbb{Z}_p)$. In this article we prove that, if $\mathbb{G}_1^{\circ}$ is the smallest closed normal subgroup of $\mathbb{G}_1^{\circ}$ which contains $\mathbb{G}_2^{\circ}$ and $\Gamma_2\curvearrowright G_2$ has spectral gap, then $\Gamma_1\curvearrowright G_1$ has spectral gap.