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Counting and effective rigidity in algebra and geometry

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Abstract

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.

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Acknowledgements

The authors thank Jayadev Athreya, Richard Canary, Ted Chinburg, Britain Cox, Peter Doyle, Daniel Fiorilli, Tsachik Gelander, Grant Lakeland, Chris Leininger, Jeff Meyer, Nick Miller, Gopal Prasad, Alan Reid, and Matthew Stover for conversations on the material in this article. We also thank the anonymous referee for detailed comments on an earlier version that corrected a mistake in Theorem 1.1 and helped improve the exposition. BL was partially supported by NSF RTG Grant DMS-1045119 and an NSF Mathematical Sciences Postdoctoral Fellowship. DBM was partially supported by NSF Grant DMS-1105710. PP was partially supported by NSF Grant DMS-1402268. LT was partially supported NSF VIGRE Grant DMS-0738586 and by an AMS Simons Travel Grant.

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Authors and Affiliations

  1. Department of Mathematics, Oberlin College, Oberlin, OH, 44074, USA

    Benjamin Linowitz & Lola Thompson

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    D. B. McReynolds

  3. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Paul Pollack

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  1. Benjamin Linowitz
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Linowitz, B., McReynolds, D.B., Pollack, P. et al. Counting and effective rigidity in algebra and geometry. Invent. math. 213, 697–758 (2018). https://doi.org/10.1007/s00222-018-0796-y

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