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Relatively hyperbolic groups with fixed peripherals

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Abstract

We build quasi-isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors.

We prove that, given any finite collection of finitely generated groups H each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi-isometry types of one-ended groups which are hyperbolic relative to H.

The groups are constructed using classical small cancellation theory over free products.

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References

  1. E. Alibegović, A combination theorem for relatively hyperbolic groups, Bulletin of the London Mathematical Society 37 (2005), 459–466.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Behrstock, C. Druţu and L. Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Mathematische Annalen 344 (2009), 543–595.

    Article  MathSciNet  MATH  Google Scholar 

  3. G. C. Bell and A. N. Dranishnikov, Asymptotic dimension, Topology and its Applications 155 (2008), 1265–1296.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. C. Bell, A. N. Dranishnikov and J. E. Keesling, On a formula for the asymptotic dimension of free products, Fundamenta Mathematicae 183 (2004), 39–45.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. C. Bell and K. Fujiwara, The asymptotic dimension of a curve graph is finite, Journal of the London Mathematical Society 77 (2008), 33–50.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bestvina, K. Bromberg and K. Fujiwara, Constructing group actions on quasitrees and applications to mapping class groups, Publications Mathématiques. Institut de Hautes Études Scientifiques 122 (2015), 1–64.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Borel and G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, Journal für die Reine und Angewandte Mathematik 298 (1978), 53–64.

    MathSciNet  MATH  Google Scholar 

  8. B. H. Bowditch, Relatively hyperbolic groups, International Journal of Algebra and Computation 22 (2012), 1250016.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Champetier, Propriétés statistiques des groupes de présentation finie, Advances in Mathematics 116 (1995), 197–262.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Cordes and D. Hume, Stability and the Morse boundary, Journal of the London Mathematical Society 95 (2017), 963–988.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Dahmani, Combination of convergence groups, Geometry and Topology 7 (2003), 933–963.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Dahmani, V. Guirardel and P. Przytycki, Random groups do not split, Mathematische Annalen 349 (2011), 657–673.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Druţu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), 959–1058.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Druţu, Relatively hyperbolic groups: geometry and quasi-isometric invariance, Commentarii Mathematici Helvetici 84 (2009), 503–546.

    MathSciNet  MATH  Google Scholar 

  15. M. J. Dunwoody, The accessibility of finitely presented groups, Inventiones Mathematicae 81 (1985), 449–458.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. G. Durham and S. J. Taylor, Convex cocompactness and stability in mapping class groups, Algebraic & Geometric Topology 15 (2015), 2839–2859.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Farb, Relatively hyperbolic groups, Geometric and Functional Analysis 8 (1998), 810–840.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. W. Groff, Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups, Journal of Topology and Analysis 5 (2013), 451–475.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Gromov, Hyperbolic groups, in Essays in Group Theory, Mathematical Sciences Research Institute Publications, Vol. 8, Springer, New York, 1987, pp. 75–263.

  20. D. Gruber, Personal communication.

  21. D. Hume, Embedding mapping class groups into a finite product of trees, Groups, Geometry, and Dynamics 11 (2017), 613–647.

    Article  MathSciNet  MATH  Google Scholar 

  22. O. Kharlampovich and A. G. Myasnikov, Equations and fully residually free groups, in Combinatorial and Geometric Group Theory, Trens in Mathematics, Birkhäuser/Springer, Basel, 2010, pp. 203–242.

    Google Scholar 

  23. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

    Book  MATH  Google Scholar 

  24. J. M. Mackay, Conformal dimension and random groups, Geometric and Functional Analysis 22 (2012), 213–239.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. M. Mackay and A. Sisto, Embedding relatively hyperbolic groups in products of trees, Algebraic & Geometric Topology 13 (2013), 2261–2282.

    Article  MathSciNet  MATH  Google Scholar 

  26. D. V. Osin, Asymptotic dimension of relatively hyperbolic groups, International Mathematics Research Notices 35 (2005), 2143–2161.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Memoirs of the American Mathematical Society 179 (2006).

    Google Scholar 

  28. P. Papasoglu and K. Whyte, Quasi-isometries between groups with infinitely many ends, Commentarii Mathematici Helvetici 77 (2002), 133–144.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Roe, Hyperbolic groups have finite asymptotic dimension, Proceedings of the American Mathematical Society 133 (2005), 2489–2490.

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Schwartz, The quasi-isometry classification of rank one lattices, Publications Mathématiques. Institut des Hautes Etudes Scientifiques 82 (1995), 133–168.

    Article  MathSciNet  MATH  Google Scholar 

  31. Z. Sela, Diophantine geometry over groups. I. Makanin–Razborov diagrams, Publications Mathématiques. Institut des Hautes Études Scientifiques 93 (2001), 31–105.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Silberman, Addendum to: “Random walk in random groups” [Geom. Funct. Anal. 13 (2003), no. 1, 73–146; MR1978492] by M. Gromov, Geometric and Functional Analysis 13 (2003), 147–177.

    Article  MathSciNet  Google Scholar 

  33. A. Sisto, Projections and relative hyperbolicity, L’Enseignement Mathématique 59 (2013), 165–181.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Stallings, Groups of dimension 1 are locally free, Bulletin of the American Mathematical Society 74 (1968), 361–364.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Stallings, Group Theory and Three-Dimensional Manifolds, Yale Mathematical Monographs, Vol. 4, Yale University Press, New Haven, CT, 1971.

  36. R. Strebel, Appendix. Small cancellation groups, in Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988), Progress in Mathematics, Vol. 83, Birkhäuser Boston, Boston, MA, 1990, pp. 227–273.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland

    Matthew Cordes

  2. Matheamtical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK

    David Hume

Authors
  1. Matthew Cordes
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  2. David Hume
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Correspondence to David Hume.

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Cordes, M., Hume, D. Relatively hyperbolic groups with fixed peripherals. Isr. J. Math. 230, 443–470 (2019). https://doi.org/10.1007/s11856-019-1830-5

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  • DOI: https://doi.org/10.1007/s11856-019-1830-5

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