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High-dimensional expansion and soficity of groups

Lukas Gohla, Andreas Thom

Abstract

For $d \geq 4$ and $p$ a sufficiently large prime, we construct a lattice $Γ\leq {\rm PSp}_{2d}(\mathbb Q_p),$ such that its universal central extension cannot be sofic if $Γ$ satisfies some weak form of stability in permutations. In the proof, we make use of high-dimensional expansion phenomena and, extending results of Lubotzky, we construct new examples of cosystolic expanders over arbitrary finite abelian groups.

High-dimensional expansion and soficity of groups

Abstract

For and a sufficiently large prime, we construct a lattice such that its universal central extension cannot be sofic if satisfies some weak form of stability in permutations. In the proof, we make use of high-dimensional expansion phenomena and, extending results of Lubotzky, we construct new examples of cosystolic expanders over arbitrary finite abelian groups.
Paper Structure (14 sections, 20 theorems, 45 equations)

This paper contains 14 sections, 20 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Group cohomology and applications
  3. Measured Boolean algebras
  4. Metric group cohomology
  5. Shapiro's lemma in the metric context
  6. Bruhat-Tits buildings
  7. Cosystolic inequalities and expansion
  8. An application to weak containment
  9. An application to residually finite groups
  10. Approximation and stability
  11. Sofic approximations
  12. Stability
  13. Proof of the main result
  14. A candidate for a non-sofic group

Key Result

Lemma 2.5

Let $\Lambda \leq \Gamma$ be a subgroup. There is a natural isometric isomorphism such that for all $[c] \in {\rm H}^i(B\Gamma,A)$.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: Shapiro
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 34 more