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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


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Volume 2019, Issue 757

Issues

Systolic geometry and simplicial complexity for groups

Ivan Babenko
  • Institut Montpelliérain Alexander Grothendieck, Bat. 9, Université Montpellier 2, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
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/ Florent Balacheff
  • Laboratoire Paul Painlevé, Bat. M2, Université de Lille – Sciences et Technologies, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
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/ Guillaume Bulteau
  • Institut Montpelliérain Alexander Grothendieck, Bat. 9, Université Montpellier 2, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
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Published Online: 2017-11-12 | DOI: https://doi.org/10.1515/crelle-2017-0041

Abstract

Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called simplicial complexity that allows to obtain a quite satisfactory answer to his question. Using this new complexity, we also derive new results on systolic area for groups that specify its topological behaviour.

References

  • [1]

    I. Babenko and F. Balacheff, Géométrie systolique des sommes connexes et des revêtements cycliques, Math. Ann. 333 (2005), 157–180. CrossrefGoogle Scholar

  • [2]

    I. Babenko and F. Balacheff, Systolic volume of homology classes, Algebr. Geom. Topol. 15 (2015), 733–767. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    F. Balacheff, H. Parlier and S. Sabourau, Short loop decompositions of surfaces, Geom. Funct. Anal. 22 (2012), 37–73. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    G. Bulteau, Les groupes de petite complexité simpliciale, preprint (2015), https://hal.archives-ouvertes.fr/hal-01168493.

  • [5]

    P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane, Invent. Math. 117 (1994), 27–56. CrossrefGoogle Scholar

  • [6]

    J. C. Cha, Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations, preprint (2015), https://arxiv.org/abs/1506.00757.

  • [7]

    J. C. Cha, Complexity of surgery manifolds and Cheeger–Gromov invariants, Int. Math. Res. Not. IMRN (2016), no. 18, 5603–5615. Google Scholar

  • [8]

    T. Delzant, Décomposition d’un groupe en produit libre ou somme amalgamée, J. reine angew. Math. 470 (1996), 153–180. Google Scholar

  • [9]

    M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. CrossrefGoogle Scholar

  • [10]

    M. Gromov, Systoles and intersystolic inequalities, Actes de la table ronde de géométrie différentielle (Luminy 1992), Sémin. Congr. 1, Société Mathématique de France, Paris (1996), 291–362. Google Scholar

  • [11]

    A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge 2002. Google Scholar

  • [12]

    W. Jaco, H. Rubinstein and S. Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009), 157–180. CrossrefWeb of ScienceGoogle Scholar

  • [13]

    W. Jaco, H. Rubinstein and S. Tillmann, Coverings and minimal triangulations of 3-manifolds, Algebr. Geom. Topol. 11 (2011), 1257–1265. CrossrefWeb of ScienceGoogle Scholar

  • [14]

    W. Jaco, H. Rubinstein and S. Tillmann, 2-Thurston norm and complexity of 3-manifolds, Math. Ann. 356 (2013), 1–22. Google Scholar

  • [15]

    M. Jungerman and G. Ringel, Minimal triangulations on orientable surfaces, Acta Math. 145 (1980), 121–154. CrossrefGoogle Scholar

  • [16]

    I. Kapovich and P. Schupp, Delzant’s T-invariant, one-relator groups, and Kolmogorov’s complexity, Comment. Math. Helv. 80 (2005), 911–933. Google Scholar

  • [17]

    A. G. Kurosh, The theory of groups, Chelsea Publishing, New York 1960. Google Scholar

  • [18]

    W. S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, New York 1967. Google Scholar

  • [19]

    S. Matveev, Algorithmic topology and classification of 3-manifolds, 2nd ed., Algorithms Comput. Math. 9, Springer, Berlin 2007. Google Scholar

  • [20]

    S. Matveev and E. Pervova, Lower bounds for the complexity of three-dimensional manifolds, Dokl. Akad. Nauk. 378 (2001), 151–152. Google Scholar

  • [21]

    S. Matveev, C. Petronio and A. Vesnin, Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds, J. Aust. Math. Soc. 86 (2009), 205–219. Web of ScienceCrossrefGoogle Scholar

  • [22]

    J. Milnor, Groups which act on Sn without fixed points, Amer. J. Math. 79 (1957), 623–630. Google Scholar

  • [23]

    E. Pervova and C. Petronio, Complexity and T-invariant of Abelian and Milnor Groups, and complexity of 3-manifolds, Math. Nachr. 281 (2008), 1182–1195. CrossrefWeb of ScienceGoogle Scholar

  • [24]

    P. Pu, Some inequalities in certain non-orientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71. CrossrefGoogle Scholar

  • [25]

    Y. Rudyak and S. Sabourau, Systolic invariants of groups and 2-complexes via Grushko decomposition, Ann. Inst. Fourier 58 (2008), 777–800. Web of ScienceCrossrefGoogle Scholar

  • [26]

    H. Seifert and W. Threlfall, A textbook of topology, Academic Press, New York 1980. Google Scholar

About the article

Received: 2016-12-12

Revised: 2017-08-27

Published Online: 2017-11-12

Published in Print: 2019-12-01


Funding Source: Russian Science Foundation

Award identifier / Grant number: 10-01-00257-a

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: Finsler-12-BS01-0009-02

This work was partially supported by the grant RFSF 10-01-00257-a and the program ANR Finsler-12-BS01-0009-02.


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 757, Pages 247–277, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2017-0041.

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