Abstract
In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and \(\delta \)-hyperbolic. In this case, we also show that the mapping class group acts on the grand arc graph by isometries and acts on the visible boundary continuously. When the surface has stable maximal ends, we also show that this action has finitely many orbits.
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Notes
This fact was communicated to the first author by Kasra Rafi, and follows from a construction of a surface with a single, unstable maximal end.
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Acknowledgements
We would like to thank Mahan Mj, Kasra Rafi and Ferrán Valdez for helpful conversations. We would like to thank Tyrone Ghaswala for comments and questions on an earlier draft. We would also like to thank the anonymous referee for providing helpful comments. The first author was partially supported by an NSERC-PGSD Fellowship, a Queen Elizabeth II Scholarship, and a FAST Scholarship at the University of Toronto. The second author was supported by the National Science Foundation under Grant No. DMS-1928930 while participating in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester. The second author was also partially supported by an NSERC-PDF Fellowship.
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Bar-Natan, A., Verberne, Y. The grand arc graph. Math. Z. 305, 20 (2023). https://doi.org/10.1007/s00209-023-03337-z
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DOI: https://doi.org/10.1007/s00209-023-03337-z