Abstract
A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, CAT(0) groups with rank 1 elements, mapping class groups and Out(F n ). As an application, we show that mapping class groups act on finite products of δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.
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The first two authors gratefully acknowledge the support by the National Science Foundation. The third author is supported in part by Grant-in-Aid for Scientific Research (No. 23244005).
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Bestvina, M., Bromberg, K. & Fujiwara, K. Constructing group actions on quasi-trees and applications to mapping class groups. Publ.math.IHES 122, 1–64 (2015). https://doi.org/10.1007/s10240-014-0067-4
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DOI: https://doi.org/10.1007/s10240-014-0067-4