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Transactions of the American Mathematical Society

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A metrizable topology on the contracting boundary of a group


Authors: Christopher H. Cashen and John M. Mackay
Journal: Trans. Amer. Math. Soc. 372 (2019), 1555-1600
MSC (2010): Primary 20F65, 20F67
DOI: https://doi.org/10.1090/tran/7544
Published electronically: May 7, 2019
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Abstract: The `contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.


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Additional Information

Christopher H. Cashen
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

John M. Mackay
Affiliation: School of Mathematics, University of Bristol, Bristol, United Kingdom

DOI: https://doi.org/10.1090/tran/7544
Keywords: Contracting boundary, Morse boundary, boundary at infinity, contracting geodesic, divagation
Received by editor(s): August 18, 2017
Received by editor(s) in revised form: February 13, 2018
Published electronically: May 7, 2019
Additional Notes: The first author thanks the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Non-positive curvature: group actions and cohomology where work on this paper was undertaken. This work was partially supported by EPSRC Grant Number EP/K032208/1 and by the Austrian Science Fund (FWF): P30487-N35.
The second author was supported in part by the National Science Foundation under Grant DMS-1440140 while visiting the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester, and in part by EPSRC grant EP/P010245/1.
Article copyright: © Copyright 2019 American Mathematical Society