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

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 ${\mathbb{F}}_2⋉{\mathbb{Z}}^2$. We also obtain an upper bound for the Kazhdan constants of pairs of orderable groups, depending only on the cardinal of the generating subset.