Stability, approximable quotients, and higher property (T)

TL;DR

This paper constructs finitely presented groups with property (T) and Frobenius stability that are significantly different from having property (T$_2$), demonstrating that property (T$_2$) does not pass to quotients.

Contribution

It introduces new groups with property (T) and Frobenius stability that differ from property (T$_2$), and shows property (T$_2$) is not preserved under quotients.

Findings

Existence of finitely presented Frobenius stable groups with property (T) but not property (T$_2$)

Property (T$_2$) does not pass to quotients

Construction method for such groups

Abstract

We construct a wealth of groups that are finitely presented, Frobenius stable, have property (T), but are very far from having property (T$_2$). Our method also shows that property (T$_2$) does not pass to quotients.