Let $\Gamma _2\subseteq \Gamma _1$ be finitely generated subgroups of ${\operatorname{GL}}_{n_0}({\mathbb{Z}}[1/q_0])$ where $q_0$ is a positive integer. For $i=1$ or $2$, let ${\mathbb{G}}_i$ be the Zariski-closure of $\Gamma _i$ in $({\operatorname{GL}}_{n_0})_{{\mathbb{Q}}}$, ${\mathbb{G}}_i^{\circ }$ be the Zariski-connected component of ${\mathbb{G}}_i$, and let $G_i$ be the closure of $\Gamma _i$ in $\prod _{p\nmid q_0}{\operatorname{GL}}_{n_0}({\mathbb{Z}}_p)$. In this article we prove that if ${\mathbb{G}}_1^{\circ }$ is the smallest closed normal subgroup of ${\mathbb{G}}_1^{\circ }$ that contains ${\mathbb{G}}_2^{\circ }$ and $\Gamma _2\curvearrowright G_2$ has spectral gap, then $\Gamma _1\curvearrowright G_1$ has spectral gap.
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Morphisms of Character Varieties
Let $k$ be a field, let $H \subset G$ be (possibly disconnected) reductive groups over $k$, and let $\Gamma $ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $k$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use BruhatâTits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings.
more » « less- Award ID(s):
- 1840234
- PAR ID:
- 10531675
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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