Property (T) and Many Quotients

TL;DR

This paper demonstrates that certain free algebras over rich operads have large subgroups of automorphisms with Kazhdan's property (T), leading to groups with property (T) mapping onto large powers of alternating groups.

Contribution

It establishes the presence of Kazhdan's property (T) in automorphism groups of free algebras over rich operads, linking algebraic structures to property (T).

Findings

Large subgroups of automorphisms have property (T)

Existence of groups with property (T) mapping onto large powers of alternating groups

Connections between operad-based free algebras and property (T) groups

Abstract

We prove that, for the free algebra over a sufficiently rich operad, a large subgroup of its group of tame automorphisms has Kazhdan's property (T). We deduce that there exists a group with property (T) that maps onto large powers of alternating groups.