Higher property T and below-rank phenomena of lattices

TL;DR

This paper investigates higher property T in groups and lattices, offering new operator-algebraic characterizations and connecting these properties to cohomological, rigidity, and geometric phenomena in semisimple Lie groups.

Contribution

It introduces novel operator-algebraic characterizations of higher property T and proposes a unifying conjectural framework linking various phenomena in lattices.

Findings

New operator-algebraic characterizations of higher property T

Relationships between higher property T and geometric phenomena

A conjectural framework unifying these aspects

Abstract

The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances.