Kazhdan quotients of Golod-Shafarevich groups

TL;DR

This paper proves that every Golod-Shafarevich group has an infinite quotient with Kazhdan's property (T), confirming their non-amenability and advancing understanding of their structural properties.

Contribution

It establishes that all Golod-Shafarevich groups possess an infinite Kazhdan (T) quotient, a novel result linking these groups to property (T).

Findings

Golod-Shafarevich groups have infinite quotients with property (T)

Affirmative answer to non-amenability of Golod-Shafarevich groups

Advances understanding of the structure of Golod-Shafarevich groups

Abstract

The main goal of this paper is to prove that every Golod-Shafarevich group has an infinite quotient with Kazhdan's property $(T)$. In particular, this gives an affirmative answer to the well-known question about non-amenability of Golod-Shafarevich groups.