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Topology and Geometry of Random 2-Dimensional Hypertrees

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Abstract

A hypertree, or \(\mathbb {Q}\)-acyclic complex, is a higher-dimensional analogue of a tree. We study random 2-dimensional hypertrees according to the determinantal measure suggested by Lyons. We are especially interested in their topological and geometric properties. We show that with high probability, a random 2-dimensional hypertree T is aspherical, i.e., that it has a contractible universal cover. We also show that with high probability the fundamental group \(\pi _1(T)\) is hyperbolic and has cohomological dimension 2.

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Acknowledgements

M.K. is grateful to Nati Linial for suggesting the study of random hypertrees and for encouragement. We thank TU Berlin for hosting us during the 2019–2020 academic year. We also thank Russell Lyons for helpful comments on an earlier draft.

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Authors and Affiliations

  1. The Ohio State University, 100 Math Tower, 231 W 18th Ave, Columbus, OH, 43210, USA

    Matthew Kahle

  2. Chair of Discrete Mathematics / Geometry, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Andrew Newman

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  1. Matthew Kahle
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  2. Andrew Newman
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Correspondence to Matthew Kahle.

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Editor in Charge: János Pach

M.K. was supported in part by NSF-CCF Grants #1740761 and #1839358. He is grateful to the Simons Foundation for a Simons Fellowship, and to the Berlin Mathematical School for a Mercator Fellowship. A.N. was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Graduiertenkolleg 2434 “Facets of Complexity”

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Kahle, M., Newman, A. Topology and Geometry of Random 2-Dimensional Hypertrees. Discrete Comput Geom 67, 1229–1244 (2022). https://doi.org/10.1007/s00454-021-00352-x

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