Lattices in amenable groups

Bader, Uri;Caprace, Pierre-Emmanuel;Gelander, Tsachik;Mozes, Shahar
(2018) Fundamenta Mathematicae — Vol. 246, n° 3, p. 217-255 (2019)

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Authors
  • Bader, UriWeizmann Inst. of Sc.
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  • Caprace, Pierre-Emmanuelorcid-logoUCLouvain
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  • Gelander, TsachikWeizmann Inst. of Sc.
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  • Mozes, ShaharHebrew University of Jerusalem
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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 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.
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Bader, U., Caprace, P.-E., Gelander, T., & Mozes, S. (2018). Lattices in amenable groups. Fundamenta Mathematicae, 246(3), 217-255. https://hdl.handle.net/2078.5/28135 (Original work published 2019)