Lattices in amenable groups

Uri Bader, Pierre Emmanuel Caprace, Tsachik Gelander, Shahar Mozes

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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 that 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.

Original languageAmerican English
Pages (from-to)217-255
Number of pages39
JournalFundamenta Mathematicae
Issue number3
StatePublished - 2019

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2019


  • Amenable groups
  • Discrete subgroups
  • Lattices


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