Publicacions Matemàtiques

Asymptotically CAT(0) Groups

Aditi Kar

Full-text: Open access

Abstract

We develop a general theory for asymptotically $\operatorname{CAT}(0)$ groups; these are groups acting geometrically on a geodesic space, all of whose asymptotic cones are $\operatorname{CAT}(0)$.

Article information

Source
Publ. Mat., Volume 55, Number 1 (2011), 67-91.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.pm/1298670084

Mathematical Reviews number (MathSciNet)
MR2779576

Zentralblatt MATH identifier
1271.20057

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
Asymptotically CAT(0) groups asymptotic cones quasi-isometries graphs of groups finiteness properties Riemannian metrics

Citation

Kar, Aditi. Asymptotically CAT(0) Groups. Publ. Mat. 55 (2011), no. 1, 67--91. https://projecteuclid.org/euclid.pm/1298670084


Export citation

References

  • B. H. Bowditch, Relatively hyperbolic groups, Preprint (1999).
  • M. R. Bridson and A. Haefliger, “Metric spaces of non-positive curvature”, Grundlehren der Mathematischen Wissenschaften 319, Springer-Verlag, Berlin, 1999.
  • T. Delzant and M. Gromov, Courbure mésoscopique et théorie de la toute petite simplification, J. Topol. 1(4) (2008), 804\Ndash836.
  • C. Druţu, Quasi-isometry invariants and asymptotic cones, International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000), Internat. J. Algebra Comput. 12(1–2) (2002), 99\Ndash135.
  • C. Druţu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, With an appendix by Denis Osin and Sapir, Topology 44(5) (2005), 959\Ndash1058.
  • T. Elsner, Systolic groups with isolated flats, unpublished.
  • B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8(5) (1998), 810\Ndash840.
  • S. M. Gersten, Isoperimetric and isodiametric functions of finite presentations, in: “Geometric group theory”, Vol. 1 (Sussex, 1991), London Math. Soc. Lecture Note Ser. 181, Cambridge Univ. Press, Cambridge, 1993, pp. 79\Ndash96.
  • M. Gromov, Hyperbolic groups, in: “Essays in group theory”, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, pp. 75\Ndash263.
  • M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53\Ndash73.
  • M. Gromov, Mesoscopic curvature and hyperbolicity, in: “Global differential geometry: the mathematical legacy of Alfred Gray” (Bilbao, 2000), Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001, pp. 58\Ndash69.
  • D. Groves, Limit groups for relatively hyperbolic groups. I. The basic tools, Algebr. Geom. Topol. 9(3) (2009), 1423\Ndash1466.
  • T. Januszkiewicz and J. Świątkowski, Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 1\Ndash85.
  • A. Kar, Discrete groups and CAT(0) asymptotic cones, Thesis (Ph.D.), The Ohio State University (2008).
  • G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikov conjecture, Ann. of Math. (2) 158(1) (2003), 165\Ndash206.
  • A. Yu. Ol'shanskii, D. V. Osin, and M. V. Sapir, Lacunary hyperbolic groups, With an appendix by Michael Kapovich and Bruce Kleiner, Geom. Topol. 13(4) (2009), 2051\Ndash2140.
  • J. Pach, R. Pollack, and J. Spencer, Graph distance and Euclidean distance on the grid, in: “Topics in combinatorics and graph theory” (Oberwolfach, 1990), Physica, Heidelberg, 1990, pp. 555\Ndash559.
  • S. D. Pauls, The large scale geometry of nilpotent Lie groups, Comm. Anal. Geom. 9(5) (2001), 951\Ndash982.
  • T. Riley, Asymptotic invariants of infinite discrete groups, Thesis, University of Oxford (2002).
  • S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, I, Tôhoku Math. J. (2) 10 (1958), 338\Ndash354; II, Tôhoku Math. J. 14 (1962), 146\Ndash155.
  • S. Sasaki, Geodesics on the tangent sphere bundles over space forms, J. Reine Angew. Math. 288 (1976), 106\Ndash120.
  • J.-P. Serre, Arbres, amalgames, ${\rm SL}_{2}$, Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque 46, Société Mathématique de France, Paris, 1977.
  • S. Thomas and B. Velickovic, Asymptotic cones of finitely generated groups, Bull. London Math. Soc. 32(2) (2000), 203\Ndash208.