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A Cup Product Obstruction to Frobenius Stability

Forrest Glebe

Abstract

A countable discrete group $Γ$ is said to be Frobenius stable if a function from the group that is "almost multiplicative" in the point Frobenius norm topology is "close" to a genuine unitary representation in the same topology. The purpose of this paper is to show that if $Γ$ is finitely generated and a non-torsion element of $H^2(Γ;\mathbb{Z})$ can be written as a cup product of two elements in $H^1(Γ;\mathbb{Z})$ then $Γ$ is not Frobenius stable. In general, 2-cohomology does not obstruct Frobenius stability. Some examples are discussed, including Thompson's group $F$ and Houghton's group $H_3$. The argument is sufficiently general to show that the same condition implies non-stability in unnormalized Schatten $p$-norms for $1<p\le\infty$.

A Cup Product Obstruction to Frobenius Stability

Abstract

A countable discrete group is said to be Frobenius stable if a function from the group that is "almost multiplicative" in the point Frobenius norm topology is "close" to a genuine unitary representation in the same topology. The purpose of this paper is to show that if is finitely generated and a non-torsion element of can be written as a cup product of two elements in then is not Frobenius stable. In general, 2-cohomology does not obstruct Frobenius stability. Some examples are discussed, including Thompson's group and Houghton's group . The argument is sufficiently general to show that the same condition implies non-stability in unnormalized Schatten -norms for .
Paper Structure (4 sections, 11 theorems, 28 equations)

This paper contains 4 sections, 11 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proofs
  4. Examples

Key Result

Theorem 1.1

Let $\Gamma$ be a finitely generated discrete group, and let $1<p\le\infty$. If there are $\alpha,\beta\in H^1(\Gamma;\mathbb{Z})$ so that $\alpha\smile\beta\in H^2(\Gamma;\mathbb{Z})$ is non-torsion then $\Gamma$ is not stable in the unnormalized Schatten $p$-norm.

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6: constructive Definition 3.3
  • Theorem 2.7
  • Lemma 3.1
  • ...and 22 more