Uniformly Lipschitzian group actions on hyperconvex spaces

Abstract

Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left\{T_{a}x:a\in G\right\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points $Fix \, G$ is a nonempty H\"older continuous retract of $M$. As a consequence, it follows that all surjective isometries acting on a bounded hyperconvex space have a common fixed point. A fixed point theorem for $L$-Lipschitzian involutions and some generalizations to the case of $\lambda$-hyperconvex spaces are also given.