A Cup Product Obstruction to Frobenius Stability

TL;DR

This paper demonstrates that certain cohomological cup product conditions in finitely generated groups serve as obstructions to Frobenius stability, with implications for Schatten norms and specific groups like Thompson's and Houghton's.

Contribution

It introduces a cohomological criterion involving cup products that obstructs Frobenius stability in finitely generated groups.

Findings

Non-torsion cup products in H^2 obstruct Frobenius stability.

The obstruction extends to Schatten p-norms for 1<p≤∞.

Examples include Thompson's group F and Houghton's H_3.

Abstract

A countable discrete group $\Gamma$ 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 $\Gamma$ is finitely generated and a non-torsion element of $H^2(\Gamma;\mathbb{Z})$ can be written as a cup product of two elements in $H^1(\Gamma;\mathbb{Z})$ then $\Gamma$ 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 $1<p\le\infty$.