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On super-rigidity of Gromov's random monster group

Kajal Das

TL;DR

This work establishes super-rigidity for Gromov's random monster groups $\Gamma_\alpha$, showing that any homomorphism $\phi_{\alpha}: \Gamma_{\alpha} \to G$ has finite image for almost every $\alpha$ when $G$ ranges over several large and structurally rich classes, including mapping class groups, braid groups, $Out(F_N)$, $Aut(F_N)$, hierarchically hyperbolic groups, a-$L^p$-menable, and K-amenable groups. It introduces hereditary super-rigidity and proves it for $\Gamma_\alpha$ with respect to a-$L^p$-menable and K-amenable targets, respectively, and it proves a stability theorem showing how these rigidity properties behave under extensions. The proofs combine the random-monster construction (and its property (T) and non-embeddability) with dynamics on hyperbolic or hierarchically hyperbolic spaces, leveraging short exact sequences and structural results for the target groups. The results yield concrete consequences for actions of $\Gamma_\alpha$ on MCG curve complexes, $FF_N$, and related spaces, and establish a framework for extending rigidity phenomena to broader classes via hereditary SR and stability. Overall, the paper advances the understanding of rigidity phenomena in random groups and their interactions with geometrically or analytically rich target groups, highlighting potential avenues for rigidity in higher-rank or biautomatic settings.

Abstract

In this article, we show super-rigidity of Gromov's random monster group. We prove that any morphism $φ_α$ from Gromov's random monster group $Γ_α$ to the group $G$ has finite image for almost all $α$, where $G$ is any of the following types of groups: mapping class group $MCG(S_{g,b})$, braid group $B_n$, outer automorphism group of a free group $Out(F_N)$, automorphism group of a free group $Aut(F_N)$, hierarchically hyperbolic group, a-$L^p$-menable group or K-amenable group. We introduce another property called hereditary super-rigidity and prove that $Γ_α$ has hereditary super-rigidity with respect to an a-$L^p$-menable group or a K-amenable group. We also establish a stability theorem for the groups with respect to which $Γ_α$ has super-rigidity and hereditary super-rigidity.

On super-rigidity of Gromov's random monster group

TL;DR

This work establishes super-rigidity for Gromov's random monster groups , showing that any homomorphism has finite image for almost every when ranges over several large and structurally rich classes, including mapping class groups, braid groups, , , hierarchically hyperbolic groups, a--menable, and K-amenable groups. It introduces hereditary super-rigidity and proves it for with respect to a--menable and K-amenable targets, respectively, and it proves a stability theorem showing how these rigidity properties behave under extensions. The proofs combine the random-monster construction (and its property (T) and non-embeddability) with dynamics on hyperbolic or hierarchically hyperbolic spaces, leveraging short exact sequences and structural results for the target groups. The results yield concrete consequences for actions of on MCG curve complexes, , and related spaces, and establish a framework for extending rigidity phenomena to broader classes via hereditary SR and stability. Overall, the paper advances the understanding of rigidity phenomena in random groups and their interactions with geometrically or analytically rich target groups, highlighting potential avenues for rigidity in higher-rank or biautomatic settings.

Abstract

In this article, we show super-rigidity of Gromov's random monster group. We prove that any morphism from Gromov's random monster group to the group has finite image for almost all , where is any of the following types of groups: mapping class group , braid group , outer automorphism group of a free group , automorphism group of a free group , hierarchically hyperbolic group, a--menable group or K-amenable group. We introduce another property called hereditary super-rigidity and prove that has hereditary super-rigidity with respect to an a--menable group or a K-amenable group. We also establish a stability theorem for the groups with respect to which has super-rigidity and hereditary super-rigidity.
Paper Structure (18 sections, 30 theorems, 10 equations)

This paper contains 18 sections, 30 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Some definitions, notations and useful results
  4. Gromov hyperbolic space and Gromov hyperbolic group
  5. Elementary and non-elementary action
  6. Expander graphs and dg-bounded graphs
  7. Gromov's random monster group
  8. Mapping Class Group
  9. Braid group
  10. Outer automorphism group of a free group
  11. Acylindrically Hyperbolic Group
  12. Hierarchically Hyperbolic Group
  13. Groups with Property $F_{L^p}$ and a-$FL^p$-menability
  14. K-amenable groups
  15. Hereditary super-rigidity and the stable properties of super-rigidity
  16. ...and 3 more sections

Key Result

Theorem 1.1

For every $j\geq 1$ and for any morphism $\Gamma_\alpha\rightarrow MCG(S_{g,b})$, $\Gamma_\alpha$ has finite image in $MCG(S_{g,b})$ for almost every $\alpha\in \mathcal{A}(\Omega, T^j)$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 54 more