Relative Property (T) and the Vanishing of the first $\ell^2$-Betti number

TL;DR

This paper demonstrates that groups with relative property (T) often have a zero first ll^2-Betti number, with specific applications to elementary matrix groups over countable rings.

Contribution

It establishes a link between relative property (T) and the vanishing of the first ll^2-Betti number, extending results to elementary matrix groups over rings.

Findings

Certain families with relative property (T) have trivial first ll^2-Betti number.

Application to L_n(\u211b) over countable rings of characteristic 0.

Provides new insights into the structure of groups with property (T).

Abstract

In this paper, we show that certain families with relative property (T) have trivial first $\ell^2$-Betti number. We apply this to the elementary matrix group $\EL_n(\R)$ where $\R$ is any countable unital ring of characteristic 0.