A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real line

TL;DR

This paper explores the relationship between Lipschitz and Kazhdan constants in groups of homeomorphisms of the real line, providing obstructions and bounds related to group actions with property (T).

Contribution

It introduces a new obstruction criterion linking Lipschitz and Kazhdan constants for groups with Relative Property (T) acting on the real line.

Findings

Obstruction for groups with Relative Property (T) to act by bi-Lipschitz homeomorphisms.

Explicit lower bounds for Lipschitz constants in certain group actions.

Upper bounds for Kazhdan constants depending on generating set size.

Abstract

We exhibit an obstruction for groups with Relative Property (T) to act on the real line by bi-Lipschitz homeomorphisms. This condition is expressed in terms of the Lipschitz and Kazhdan constants associated to finite generating subsets. As an application, we obtain an explicit lower bound for the Lipschitz constants associated to actions of the semidirect product $\mathbb{F}_2\ltimes\mathbb{Z}^2$. We also obtain an upper bound for the Kazhdan constants of pairs of orderable groups, depending only on the cardinal of the generating subset.