Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$

TL;DR

This paper introduces the concept of almost invariant CND kernels to characterize groups with proper Lipschitz actions on subspaces of L^1, linking group actions to kernel invariance properties.

Contribution

It defines almost invariant CND kernels and characterizes groups acting properly on L^1 subspaces, extending understanding of group actions and kernel invariance.

Findings

Groups acting properly on products of quasi-trees satisfy the condition.

Weakly amenable groups with Cowling-Haagerup constant 1 satisfy the condition.

a-TTT-menable groups satisfy the condition.

Abstract

We define the notion of almost invariant conditionally negative definite kernel and use it to give a characterisation of groups admitting a proper uniformly Lipschitz affine action on a subspace of an $L^1$ space. We show that this condition is satisfied by groups acting properly on products of quasi-trees, weakly amenable groups with Cowling-Haagerup constant 1, and a-TTT-menable groups.