An alternative proof of Kazhdan property for elementary groups

TL;DR

This paper presents a simpler, alternative proof of Kazhdan's property (T) for elementary groups, avoiding estimates of Kazhdan constants, and relates to recent work on fixed point properties.

Contribution

It offers a new, more straightforward proof of Kazhdan property (T) for elementary groups, expanding understanding without relying on Kazhdan constant estimates.

Findings

Provides a simpler proof of Kazhdan property (T)

Eliminates the need for Kazhdan constant estimates

Connects to recent fixed point property results

Abstract

In 2010, Invent. Math., Ershov and Jaikin-Zapirain proved Kazhdan's property (T) for elementary groups. This expository article focuses on presenting an alternative simpler proof of that. Unlike the original one, our proof supplies no estimate of Kazhdan constants. It may be regarded as a specific example of the results in the paper "Upgrading fixed points without bounded generation" (arXiv:1505.06728, forthcoming version) by the author.