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Tame combings and easy groups

  • Daniele Ettore Otera EMAIL logo and Valentin Poénaru
Published/Copyright: August 11, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 3

Abstract

We consider finitely presented groups admitting 0-combings which are both Lipschitz (in the sense of Thurston) and tame (as defined by Mihalik and Tschantz in [8]). What we prove is that such groups are easy (and hence QSF by [11]), in the sense that they admit an easy representation (that is a map from a 2-complex to a singular 3-manifold associated to the group, satisfying several topological conditions with a strong control over singularities). Besides its own interest, one may also try to adapt the proof in a wider context, namely for groups admitting tame 1-combings (as in [8]), in order to prove the easy-representability for a larger class of finitely presented groups (note that there are still no examples of finitely presented groups which are not tame 1-combable).

Keywords: Combings; quasi-simple filtration (QSF); inverse representations; easy groups
MSC 2010: 57M05; 57M10; 57N35

Communicated by Karl Strambach

Funding source: European Science Foundation

Award Identifier / Grant number: 6028

Funding source: Lietuvos Mokslo Taryba

Award Identifier / Grant number: MIP-046/2014/LSS-580000-446

Funding statement: The first author was partially supported by the ESF short-visit grant 6028 (within the Project ‘Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics – ITGP’), by INDAM of Italy, and by the Research Council of Lithuania Grant No. MIP-046/2014/LSS-580000-446 (Researcher teams’ projects).

Acknowledgements

The authors are thankful to Gérard Besson, Louis Funar, David Gabai, Frédéric Haglund and Corrado Tanasi for helpful conversations and comments on the subject. The first author wishes to thank the Laboratoire de Mathématiques d’Orsay (Université Paris-Sud 11) for the warm hospitality.

References

[1] Bessières L., Besson G., Boileau M., Maillot S. and Porti J., Geometrisation of 3-Manifolds, EMS Tracts Math. 13, European Mathematical Society, Zürich, 2010. 10.4171/082Search in Google Scholar

[2] Brick S. G. and Mihalik M. L., The QSF property for groups and spaces, Math. Z. 220 (1995), no. 2, 207–217. 10.1007/BF02572610Search in Google Scholar

[3] Davis M. W., Groups generated by reflections and aspherical manifolds not covered by Euclidean spaces, Ann. of Math. (2) 117 (1983), 293–324. 10.2307/2007079Search in Google Scholar

[4] Epstein D., Cannon J. W., Holt D., Levy S., Paterson M. and Thurston W. P., Word Processing in Groups, Jones and Bartlett Publishers, Boston, 1992. 10.1201/9781439865699Search in Google Scholar

[5] Funar L. and Gadgil S., On the geometric simple connectivity of open manifolds, Int. Math. Res. Not. IMRN 2004 (2004), no. 24 1193–1248. 10.1155/S1073792804130195Search in Google Scholar

[6] Funar L. and Otera D. E., On the wgsc and qsf tameness conditions for finitely presented groups, Groups Geom. Dyn. 4 (2010), no. 3, 549–596. 10.4171/GGD/95Search in Google Scholar

[7] Gersten S. M. and Stallings J. R., Casson’s idea about 3-manifolds whose universal cover is ℝ3, Internat. J. Algebra Comput. 1 (1992), no. 4, 395–406. 10.1142/S0218196791000274Search in Google Scholar

[8] Mihalik M. L. and Tschantz S. T., Tame combings of groups, Trans. Amer. Math. Soc. 349 (1992), 4251–4264. 10.1090/S0002-9947-97-01772-8Search in Google Scholar

[9] Otera D. E., Topological tameness conditions of spaces and groups: Results and developments, Lith. Math. J. 56 (2016), no. 3, 357–376. 10.1007/s10986-016-9323-2Search in Google Scholar

[10] Otera D. E., On Poénaru’s inverse-representations, Quaest. Math., to appear. 10.2989/16073606.2017.1286403Search in Google Scholar

[11] Otera D. E. and Poénaru V., “Easy” representations and the qsf property for groups, Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 3, 385–398. 10.36045/bbms/1347642372Search in Google Scholar

[12] Otera D. E. and Poénaru V., Finitely presented groups and the Whitehead nightmare, Groups Geom. Dyn., to appear. 10.4171/GGD/397Search in Google Scholar

[13] Otera D. E., Poénaru V. and Tanasi C., On geometric simple connectivity, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), no. 2, 157–176. Search in Google Scholar

[14] Otera D. E. and Russo F., On the wgsc property in some classes of groups, Mediterr. J. Math. 6 (2009), no. 4, 501–508. 10.1007/s00009-009-0021-8Search in Google Scholar

[15] Otera D. E. and Russo F., On topological filtrations of groups, Period. Math. Hungar. 72 (2016), no. 2, 218–223. 10.1007/s10998-016-0129-0Search in Google Scholar

[16] Poénaru V., Killing handles of index one stably, and π1∞, Duke Math. J. 63 (1991), no. 2, 431–447. 10.1215/S0012-7094-91-06319-2Search in Google Scholar

[17] Poénaru V., On the equivalence relation forced by the singularities of a non-degenerate simplicial map, Duke Math. J. 63 (1991), no. 2, 421–429. 10.1215/S0012-7094-91-06318-0Search in Google Scholar

[18] Poénaru V., Almost convex groups, Lipschitz combing, and π1∞ for universal covering spaces of closed 3-manifolds, J. Differential Geom. 35 (1992), no. 1, 103–130. 10.4310/jdg/1214447807Search in Google Scholar

[19] Poénaru V., The collapsible pseudo-spine representation theorem, Topology 31 (1992), no. 3, 625–656. 10.1016/0040-9383(92)90055-MSearch in Google Scholar

[20] Poénaru V., Geometry “à la Gromov” for the fundamental group of a closed 3-manifold M3 and the simple connectivity at infinity of M~3, Topology 33 (1994), no. 1, 181–196. 10.1016/0040-9383(94)90041-8Search in Google Scholar

[21] Poénaru V., π1∞ and simple homotopy type in dimension 3, Low Dimensional Topology (Funchal 1998), Contemp. Math. 233, American Mathematical Society, Providence (1999), 1–28. 10.1090/conm/233/03434Search in Google Scholar

[22] Poénaru V., Discrete symmetry with compact fundamental domain and Geometric simple connectivity, preprint 2007, http://arxiv.org/abs/0711.3579. Search in Google Scholar

[23] Poénaru V., Equivariant, locally finite inverse representations with uniformly bounded zipping length, for arbitrary finitely presented groups, Geom. Dedicata 167 (2013), 91–121. 10.1007/s10711-012-9805-2Search in Google Scholar

[24] Poénaru V., All finitely presented groups are qsf, preprint 2014, http://arxiv.org/abs/1409.7325. Search in Google Scholar

[25] Poénaru V., Geometric simple connectivity and finitely presented groups, preprint 2014, http://arxiv.org/abs/1404.4283. Search in Google Scholar

[26] Poénaru V. and Tanasi C., Hausdorff combing of groups and π1∞ for universal covering spaces of closed 3-manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20 (1993), no. 3, 387–414. 10.4310/jdg/1214447807Search in Google Scholar

[27] Poénaru V. and Tanasi C., k-weakly almost convex groups and π1∞⁢M~3, Geom. Dedicata 48 (1993), 57–81. 10.1007/BF01265676Search in Google Scholar

[28] Poénaru V. and Tanasi C., Equivariant, almost-arborescent representations of open simply-connected 3-manifolds; a finiteness result, Mem. Amer. Math. Soc. 800 (2004), 1–88. 10.1090/memo/0800Search in Google Scholar

[29] Poénaru V. and Tanasi C., A group-theoretical finiteness theorem, Geom. Dedicata 137 (2008), no. 1, 1–25. 10.1007/s10711-008-9279-4Search in Google Scholar

[30] Stallings J. R., Brick’s quasi-simple filtrations for groups and 3-manifolds, Geometric Group Theory. Volume 1, London Math. Soc. Lecture Note Ser. 181, Cambridge University Press, Cambridge (1993), 188–203. 10.1017/CBO9780511661860.017Search in Google Scholar

Received: 2015-4-4
Published Online: 2016-8-11
Published in Print: 2017-5-1

© 2017 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
  3. The Selberg trace formula as a Dirichlet series
  4. On the cohomology and their torsion of real toric objects
  5. Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
  6. Non-abelian tensor and exterior products of multiplicative Lie rings
  7. On the infimum of the spectrum of a relativistic Schrödinger operator
  8. Character correspondences for symmetric groups and wreath products
  9. Univalence in locally cartesian closed ∞-categories
  10. On subordinate random walks
  11. Tame combings and easy groups
  12. On Belk’s classifying space for Thompson’s group F
  13. Subspace confinement for switched linear systems
  14. Exceptional bundles of homological dimension ${k}$
  15. Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
  16. Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
https://doi.org/10.1515/forum-2015-0063
Keywords for this article
Combings; quasi-simple filtration (QSF); inverse representations; easy groups

Articles in the same Issue

  1. Frontmatter
  2. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
  3. The Selberg trace formula as a Dirichlet series
  4. On the cohomology and their torsion of real toric objects
  5. Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
  6. Non-abelian tensor and exterior products of multiplicative Lie rings
  7. On the infimum of the spectrum of a relativistic Schrödinger operator
  8. Character correspondences for symmetric groups and wreath products
  9. Univalence in locally cartesian closed ∞-categories
  10. On subordinate random walks
  11. Tame combings and easy groups
  12. On Belk’s classifying space for Thompson’s group F
  13. Subspace confinement for switched linear systems
  14. Exceptional bundles of homological dimension ${k}$
  15. Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
  16. Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
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