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Things we can learn by considering random locally symmetric manifolds

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arxiv 2407.21208 v2 pith:IXBTAZG7 submitted 2024-07-30 math.GR

Tsachik Gelander This is my paper · ORCID
classification math.GR
keywords randomresultsmanifoldsmeasuressubgroupsapproachesconsideringfinitely
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In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of the isometry Lie group. The main goals are to prove results about deterministic groups and manifolds by considering appropriate measures. In this overview paper we describe several such results, observing the evolution process of the measures involved. Starting with a result whose proof considered finitely supported measures (more precisely, measures supported on finitely many conjugacy classes) and proceeding with results which were outcome of the successful and popular theory of IRS (invariant random subgroups). In the last couple of years the theory has expanded to SRS (stationary random subgroups) allowing to deal with a lot more problems and establish stronger results. In the last section we shall review a very recent (yet unpublished) result whose proof make use of random subgroups which are not even stationary.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Non-uniform higher-rank lattices are character rigid

    math.GR 2025-07 accept novelty 8.0 of 10

    Every irreducible non-uniform lattice in a higher-rank semisimple group of characteristic not 2 is character rigid.

  2. Hyperlinearity, stability and asymptotic spectral gap of higher rank lattices

    math.GR 2025-06 conditional novelty 7.0 of 10

    For higher-rank lattices, Hilbert-Schmidt stability implies non-hyperlinearity of certain central extensions, and character rigidity is equivalent to hyperfinite Hilbert-Schmidt stability.

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