Uniformly Lipschitzian group actions on hyperconvex spaces

Wiśnicki, Andrzej; Wośko, Jacek

doi:10.1090/proc/13016

Uniformly Lipschitzian group actions on hyperconvex spaces
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by Andrzej Wiśnicki and Jacek Wośko PDF

Proc. Amer. Math. Soc. 144 (2016), 3813-3824 Request permission

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 $\mathrm {Fix} G$ is a nonempty Hölder 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.

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