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Lattices in amenable groups

Volume 246 / 2019

Uri Bader, Pierre-Emmanuel Caprace, Tsachik Gelander, Shahar Mozes Fundamenta Mathematicae 246 (2019), 217-255 MSC: 22E40, 22D05, 22G25, 22F10, 22F30. DOI: 10.4064/fm572-9-2018 Published online: 8 April 2019

Abstract

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.

Authors

  • Uri BaderFaculty of Mathematics and Computer Science
    Weizmann Institute of Science
    Rehovot 7610001, Israel
    e-mail
  • Pierre-Emmanuel CapraceÉcole de mathématique
    Faculté des sciences
    Université catholique de Louvain
    1348 Louvain-la-Neuve, Belgique
    e-mail
  • Tsachik GelanderFaculty of Mathematics and Computer Science
    Weizmann Institute of Science
    Rehovot 7610001, Israel
    e-mail
  • Shahar MozesEinstein Institute of Mathematics
    Hebrew University of Jerusalem
    Jerusalem 9190401, Israel
    e-mail

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