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Things we can learn by considering random locally symmetric manifolds

Tsachik Gelander

TL;DR

The paper surveys probabilistic methods for studying discrete subgroups of semisimple Lie groups via invariant and stationary random subgroups (IRS and SRS). It highlights how IRS/SRS perspectives yield proofs and generalizations of fundamental results (Kazhdan–Margulis, Stuck–Zimmer), reveals that in high rank most hyperbolic manifolds are non-arithmetic, and establishes tools such as the Benjamini–Schramm topology and Betti-number convergence to connect geometry with probabilities. Key advances include confinement-to-lattice results, stiffness and Nevo–Zimmer mechanisms, and spectral-gap phenomena for product groups, which together extend classical theorems and provide robust frameworks for understanding the large-scale geometry of locally symmetric spaces. The work emphasizes the deep interplay between random subgroups, geometric topology, and representation theory, and points toward further exploration with non-stationary random subgroups and beyond-property-(T) contexts.

Abstract

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 In the last section we shall review a very recent (yet unpublished) result whose proof make use of random subgroups which are not even stationary.

Things we can learn by considering random locally symmetric manifolds

TL;DR

The paper surveys probabilistic methods for studying discrete subgroups of semisimple Lie groups via invariant and stationary random subgroups (IRS and SRS). It highlights how IRS/SRS perspectives yield proofs and generalizations of fundamental results (Kazhdan–Margulis, Stuck–Zimmer), reveals that in high rank most hyperbolic manifolds are non-arithmetic, and establishes tools such as the Benjamini–Schramm topology and Betti-number convergence to connect geometry with probabilities. Key advances include confinement-to-lattice results, stiffness and Nevo–Zimmer mechanisms, and spectral-gap phenomena for product groups, which together extend classical theorems and provide robust frameworks for understanding the large-scale geometry of locally symmetric spaces. The work emphasizes the deep interplay between random subgroups, geometric topology, and representation theory, and points toward further exploration with non-stationary random subgroups and beyond-property-(T) contexts.

Abstract

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.
Paper Structure (14 sections, 50 theorems, 72 equations, 5 figures)

This paper contains 14 sections, 50 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Most hyperbolic manifolds are non-arithmetic
  3. Invariant Random Subgroups
  4. Kazhdan Margulis theorem for IRS
  5. The Stuck--Zimmer theorem
  6. An exotic IRS in rank one
  7. The Benjamini--Schramm, the weak-* and the IRS topologies
  8. A result of the 7 Samurai
  9. Convergence of normalized Betti numbers
  10. Stationary random subgroups
  11. Confined subgroups have finite co-volume
  12. Local rigidity.
  13. Summary of the proof:
  14. Spectral gap for products and a strong version of the NST

Key Result

Theorem 1.1

Let $G$ be a simple Lie group of rank at least $2$. Let $\mu$ be an IRS in $G$ without atoms. Then a $\mu$ random subgroup is almost surely a lattice in $G$.

Figures (5)

  • Figure 1: Every $X$-manifold has a thick part.
  • Figure 2: A large volume manifold is almost everywhere fat.
  • Figure 3: The Cheeger constant of $M_n$ is too small.
  • Figure 4: Surfaces with bounded injectivity radius.
  • Figure 5: An infinite volume manifold contains contractible balls of any radius.

Theorems & Definitions (76)

  • Theorem 1.1: Stuck--Zimmer
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Definition 2.2
  • Theorem 2.3: Borel--Harish-Chandra BoHC
  • Theorem 2.4: Margulis
  • Theorem 2.5: Corlette cor, Gromov--Schoen GS
  • Example 2.7
  • Theorem 2.8: Wang Wang
  • ...and 66 more