Lattices in amenable groups
| Autor: | Pierre-Emmanuel Caprace, Shahar Mozes, Tsachik Gelander, Uri Bader |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Property (philosophy) 010102 general mathematics Amenable group Lie group Group Theory (math.GR) Locally compact group 22F30 (primary) 22D40 43A07 (secondary) 01 natural sciences Connection (mathematics) Mathematics::Group Theory Lattice (order) FOS: Mathematics Spectral gap Locally compact space 0101 mathematics Mathematics - Group Theory Mathematics |
| Zdroj: | Fundamenta Mathematicae. 246:217-255 |
| ISSN: | 1730-6329 0016-2736 |
| DOI: | 10.4064/fm572-9-2018 |
| Popis: | Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more generally nilpotent-by-nilpotent groups, do have property (M). Finally, we highlight a connection of property (M) with the subtle relation between the analytic notions of strong ergodicity and the spectral gap. Comment: 38 pages |
| Databáze: | OpenAIRE |
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