Bounding the covolume of lattices in products
| Autor: | Adrien Le Boudec, Pierre-Emmanuel Caprace |
|---|---|
| Přispěvatelé: | Université Catholique de Louvain = Catholic University of Louvain (UCL), Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), Le Boudec, Adrien, Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Normal subgroup
Pure mathematics Algebra and Number Theory Group (mathematics) 010102 general mathematics Group Theory (math.GR) 0102 computer and information sciences [MATH] Mathematics [math] 01 natural sciences Prime (order theory) [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] Mathematics::Group Theory 010201 computation theory & mathematics Simple (abstract algebra) Bounded function Product (mathematics) Totally disconnected space FOS: Mathematics Locally compact space 0101 mathematics [MATH]Mathematics [math] Mathematics - Group Theory Mathematics [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR] |
| Popis: | We study lattices in a product $G = G_1 \times \dots \times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is non-compact and every closed normal subgroup of $G_i$ is discrete or cocompact (e.g. $G_i$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\Gamma$ with dense projections is finite. The same result holds if $\Gamma$ is non-uniform, provided $G$ has Kazhdan's property (T). We show that for any compact subset $K \subset G$, the collection of discrete subgroups $\Gamma \leq G$ with $G = \Gamma K$ and dense projections is uniformly discrete, hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $Aut(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups. Comment: v1: 51 pages. v2: 52 pages. v3: final version |
| Databáze: | OpenAIRE |
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