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Filling triangulated surfaces

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Abstract

Given a triangulated closed oriented surface \((M, {\mathcal {T}}_M)\), we provide upper bounds on the number of tetrahedra needed to construct a triangulated 3-manifold \((N, {\mathcal {T}}_N)\) which bounds \((M, {\mathcal {T}}_M)\). Along the way, we develop a technique to translate (in all dimensions) between the famous Riemannian systolic inequalities of Gromov and combinatorial analogues of these inequalities.

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Acknowledgements

The authors would like to thank Dylan Thurston for some helpful comments. We would also like to thank the various anonymous referees for remarks which greatly aided in the exposition of this paper, specifically with substantially shortening the proof of Proposition 6, suggesting the addition of Proposition 8, and pointing us towards references [1, 16]. The work of the second author was partially supported by the NSF, under Grants DMS-1207782, DMS-1510640, and DMS-1812028. The research of the third author was partially supported by an AMS-Simons travel Grant.

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Authors and Affiliations

  1. Columbus, USA

    Ryan Kowalick

  2. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Jean-François Lafont

  3. Department of Mathematical and Digital Sciences, Bloomsburg University, Bloomsburg, PA, 17815, USA

    Barry Minemyer

Authors
  1. Ryan Kowalick
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  2. Jean-François Lafont
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  3. Barry Minemyer
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Correspondence to Jean-François Lafont.

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Kowalick, R., Lafont, JF. & Minemyer, B. Filling triangulated surfaces. Geom Dedicata 202, 373–386 (2019). https://doi.org/10.1007/s10711-018-00419-9

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