Abstract
We study Bowditch’s notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan’s property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock–Druţu–Sapir on the mapping class groups of orientable surfaces. We discuss a metric approximation property of finite subsets in coarse median spaces extending the classical result on approximation of Gromov hyperbolic spaces by trees.
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Notes
The tacit assumption of colourability in Bowditch’s work is not used in this lemma, since one could restrict to working with intervals in this case.
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Acknowledgments
The author would like to express his gratitude towards his Master thesis advisor, Goulnara Arzhantseva, for introducing him to this subject, sharing her knowledge, many helpful suggestions, and most of all, for encouraging him to write this paper. He wishes to thank Frédéric Paulin for giving an enlightening mini-course at the second Young Geometric Group Theory Meeting which stimulated the author’s work towards a generalization of the result concerning outer automorphisms of hyperbolic groups with Kazhdan’s property (T). The author also thanks Bogdan Nica and the anonymous referee for useful comments on the manuscript.
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The research was conducted mostly while the author prepared his Master thesis [21] at the University of Vienna and was partially supported by the European Research Council (ERC) Grant of Goulnara Arzhantseva, Grant Agreement No. 259527. The author is currently supported by the German Research Foundation (DFG) through the Research Training Group 1493 “Mathematical structures in modern quantum physics”.
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Zeidler, R. Coarse median structures and homomorphisms from Kazhdan groups. Geom Dedicata 180, 49–68 (2016). https://doi.org/10.1007/s10711-015-0090-8
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DOI: https://doi.org/10.1007/s10711-015-0090-8