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Cocycle stability in permutations of random simplicial complexes

Michael Chapman, Yuval Peled

TL;DR

The paper investigates cocycle stability for 2D Linial–Meshulam random complexes with permutation-coefficient cohomology, connecting linear stability in the 1-dimensional cocycle Cheeger constant to the existence of non-sofic hyperbolic groups. It introduces a probabilistic mid-range regime analysis, using a coupled pair (Y,Z) of complexes differing by one triangle and a stability-contradiction argument inspired by expander-code non-local testability. The work provides a_ priori bounds on the cocycle Cheeger constant across density regimes, showing a pathway from high-dimensional expansion properties to structural group-theoretic conclusions. This bridges random high-dimensional combinatorics, local-to-global spectral phenomena, and the theory of soficity, suggesting potential routes to non-sofic or non-residually finite hyperbolic groups via cocycle stability.

Abstract

Finding a non-sofic hyperbolic group will resolve two major problems in geometric group theory: Are there non sofic groups? Are there non residually finite hyperbolic groups? In this paper, we propose a new probabilistic approach to this problem, based on the cocycle stability in permutations of random 2-dimensional Linial-Meshulam complexes. Specifically, we study their cocycle stability rate, which measures how far cochains with small coboundaries are from being cocycles. Our main contribution is the following: If, in a middle triangle density range, these random complexes typically have a linear cocycle stability rate, then there exists a non-sofic hyperbolic group. Our proof method is inspired by a well known fact about the non local testability of Sipser-Spielman expander codes.

Cocycle stability in permutations of random simplicial complexes

TL;DR

The paper investigates cocycle stability for 2D Linial–Meshulam random complexes with permutation-coefficient cohomology, connecting linear stability in the 1-dimensional cocycle Cheeger constant to the existence of non-sofic hyperbolic groups. It introduces a probabilistic mid-range regime analysis, using a coupled pair (Y,Z) of complexes differing by one triangle and a stability-contradiction argument inspired by expander-code non-local testability. The work provides a_ priori bounds on the cocycle Cheeger constant across density regimes, showing a pathway from high-dimensional expansion properties to structural group-theoretic conclusions. This bridges random high-dimensional combinatorics, local-to-global spectral phenomena, and the theory of soficity, suggesting potential routes to non-sofic or non-residually finite hyperbolic groups via cocycle stability.

Abstract

Finding a non-sofic hyperbolic group will resolve two major problems in geometric group theory: Are there non sofic groups? Are there non residually finite hyperbolic groups? In this paper, we propose a new probabilistic approach to this problem, based on the cocycle stability in permutations of random 2-dimensional Linial-Meshulam complexes. Specifically, we study their cocycle stability rate, which measures how far cochains with small coboundaries are from being cocycles. Our main contribution is the following: If, in a middle triangle density range, these random complexes typically have a linear cocycle stability rate, then there exists a non-sofic hyperbolic group. Our proof method is inspired by a well known fact about the non local testability of Sipser-Spielman expander codes.

Paper Structure

This paper contains 15 sections, 13 theorems, 78 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Cocycle stability with permutation coefficients
  3. Simplicial complexes and non-commutative cohomology
  4. Cohomology with permutation coefficients
  5. Homomorphism stability and Group soficity
  6. Covering stability
  7. Local spectral expansion
  8. Structural properties of Linial--Meshulam complexes
  9. The mid range: Soficity of hyperbolic groups
  10. The sampled pair
  11. Proving Theorem \ref{['thm:midrange']}
  12. The trivial fundamental group regime
  13. The subcritical regime
  14. Strongly-connected components, collapsibility, and Cheeger constant
  15. Subcritical Linial-Meshulam Complexes

Key Result

Theorem 1.1

Let $0<\eta<1/2$, $p=n^{-1+\eta}$ and $\mathcal{Y}\sim Y(n,p)$. Then, if a.a.s. $h_1(\mathcal{Y},{\rm Sym})= \omega(n^{-3}p^{-1})$,We use the standard Bachmann--Landau asymptotic notation $O,o,\Omega,\omega$ and $\Theta$. then the fundamental group of such a complex is a.a.s. a non-sofic hyperbolic

Figures (1)

  • Figure 3.1: An $\ell$-BHK filling of $xyz$. Compare to Figure 1 in babson2011fundamental.

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.5
  • Definition 2.6
  • ...and 44 more