Uniformly Lipschitzian group actions on hyperconvex spaces

Suppose that $\{T_a:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\{T_{a}x:a\in G\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points $FixG$ 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.