On groups with Property (T_lp)

TL;DR

This paper introduces and analyzes Property (T_lp), a weaker version of Kazhdan's Property (T), for locally compact groups, exploring its characterization, properties, and implications for various classes of groups.

Contribution

It defines Property (T_lp), characterizes it for totally disconnected groups, and establishes its presence in algebraic groups, automorphism groups, and certain lattices, linking it to existing properties.

Findings

Property (T_lp) characterized by an isolation property for totally disconnected groups.

Groups with Property (T_lp) share properties with Kazhdan groups, like compact generation.

Certain algebraic and automorphism groups possess Property (T_lp).

Abstract

Let p be a real number with 1<p and different from 2. We study Property (T_lp) for a second countable locally compact group G. Property (T_lp) is a weak version of Kazhdan's Property (T), defined in terms of the orthogonal representations of G on the sequence space lp. We show that Property (T_lp) for a totally disconnected group G is characterized by an isolation property of the trivial representation from the quasi-regular representations associated to open subgroups of G. Groups with Property (T_lp) share some important properties with Kazhdan groups (compact generation, compact abelianization, ...). Simple algebraic groups over non-archimedean local fields as well as automorphism groups of regular trees have Property (T_lp). In the case of discrete groups, Property (T_lp) implies Lubotzky's Property tau and is implied by Property (F) of Glasner and Monod. We show that an irreducible lattice in a product of two locally compact groups G and H have Property (T_lp), whenever G has Property (T) and H is connected and minimally almost periodic.