Geometry & Topology

Nonpositively curved 2–complexes with isolated flats

G Christopher Hruska

Full-text: Open access

Abstract

We introduce the class of nonpositively curved 2–complexes with the Isolated Flats Property. These 2–complexes are, in a sense, hyperbolic relative to their flats. More precisely, we show that several important properties of Gromov-hyperbolic spaces hold “relative to flats” in nonpositively curved 2–complexes with the Isolated Flats Property. We introduce the Relatively Thin Triangle Property, which states roughly that the fat part of a geodesic triangle lies near a single flat. We also introduce the Relative Fellow Traveller Property, which states that pairs of quasigeodesics with common endpoints fellow travel relative to flats, in a suitable sense. The main result of this paper states that in the setting of CAT(0) 2–complexes, the Isolated Flats Property is equivalent to the Relatively Thin Triangle Property and is also equivalent to the Relative Fellow Traveller Property.

Article information

Source
Geom. Topol., Volume 8, Number 1 (2004), 205-275.

Dates
Received: 22 January 2003
Revised: 12 February 2004
Accepted: 17 December 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883366

Digital Object Identifier
doi:10.2140/gt.2004.8.205

Mathematical Reviews number (MathSciNet)
MR2033482

Zentralblatt MATH identifier
1063.20048

Subjects
Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F06: Cancellation theory; application of van Kampen diagrams [See also 57M05] 57M20: Two-dimensional complexes

Keywords
word hyperbolic nonpositive curvature thin triangles quasigeodesics isolated flats

Citation

Hruska, G Christopher. Nonpositively curved 2–complexes with isolated flats. Geom. Topol. 8 (2004), no. 1, 205--275. doi:10.2140/gt.2004.8.205. https://projecteuclid.org/euclid.gt/1513883366


Export citation

References

  • I Aitchison, Canonical flat structures on $3$-manifolds, preprint
  • J M Alonso, T Brady, D Cooper, V Ferlini, M Lustig, M Mihalik, H Short, Notes on word hyperbolic groups, (H Short, editor), from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (É Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge, NJ (1991) 3–63
  • W Ballmann, Singular spaces of nonpositive curvature, from: “Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988)”, (É Ghys, P de la Harpe, editors), Birkhäuser Boston, Boston, MA (1990) 189–201
  • W Ballmann, M Brin, Polygonal complexes and combinatorial group theory, Geom. Dedicata 50 (1994) 165–191
  • V Bangert, V Schroeder, Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4) 24 (1991) 605–634
  • N Benakli, Polygonal complexes. I. Combinatorial and geometric properties, J. Pure Appl. Algebra 97 (1994) 247–263
  • B H Bowditch, Relatively hyperbolic groups (1999), preprint, University of Southampton
  • M R Bridson, Geodesics and curvature in metric simplicial complexes, from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (É Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge, NJ (1991) 373–463
  • M R Bridson, On the existence of flat planes in spaces of nonpositive curvature, Proc. Amer. Math. Soc. 123 (1995) 223–235
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin (1999)
  • C B Croke, B Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000) 549–556
  • C B Croke, B Kleiner, The geodesic flow of a nonpositively curved graph manifold, Geom. Funct. Anal. 12 (2002) 479–545
  • M Dehn, Papers on group theory and topology, Springer-Verlag, New York (1987), translated from the German by J. Stillwell
  • P Eberlein, Geodesic flows on negatively curved manifolds. II, Trans. Amer. Math. Soc. 178 (1973) 57–82
  • V A Efromovich, E S Tihomirova, Continuation of an equimorphism to infinity, Soviet Math. Dokl. 4 (1963) 1494–1496
  • D B A Epstein, J W Cannon, D F Holt, S V F Levy, M S Paterson, W P Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA (1992)
  • B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810–840
  • S M Gersten, Reducible diagrams and equations over groups, from: “Essays in group theory”, (S M Gersten, editor), Springer, New York (1987) 15–73
  • S M Gersten, H B Short, Small cancellation theory and automatic groups, Invent. Math. 102 (1990) 305–334
  • S M Gersten, H B Short, Small cancellation theory and automatic groups. II, Invent. Math. 105 (1991) 641–662
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, (S M Gersten, editor), Springer, New York (1987) 75–263
  • F Haglund, Les polyèdres de Gromov, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991) 603–606
  • J Heber, Hyperbolische geodatische Raume, Diplomarbeit, Univ. Bonn (1987)
  • G C Hruska, Geometric invariants of spaces with isolated flats, preprint available at http://www.math.uchicago.edu/~chruska/papers
  • G C Hruska, On the relative hyperbolicity of nonpositively curved groups with isolated flats, in preparation
  • M Kapovich, B Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of $3$-manifolds, Geom. Funct. Anal. 5 (1995) 582–603
  • J Kari, P Papasoglu, Deterministic aperiodic tile sets, Geom. Funct. Anal. 9 (1999) 353–369
  • U Lang, Quasigeodesics outside horoballs, Geom. Dedicata 63 (1996) 205–215
  • R C Lyndon, On Dehn's algorithm, Math. Ann. 166 (1966) 208–228
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin (1977)
  • H A Masur, Y N Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • J P McCammond, D T Wise, Fans and ladders in small cancellation theory, Proc. London Math. Soc. (3) 84 (2002) 599–644
  • H M Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924) 25–60
  • G Moussong, Hyperbolic Coxeter groups, PhD thesis, Ohio State Univ. (1988)
  • G A Niblo, L D Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621–633
  • P Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent. Math. 121 (1995) 323–334
  • S J Pride, Star-complexes, and the dependence problems for hyperbolic complexes, Glasgow Math. J. 30 (1988) 155–170
  • D Y Rebbechi, Algorithmic properties of relatively hyperbolic groups, PhD thesis, Rutgers Univ. (2001)
  • W Rudin, Real and complex analysis, third edition, McGraw-Hill Book Co., New York (1987)
  • Z Sela, Diophantine geometry over groups: A list of research problems, available at http://www.ma.huji.ac.il/~zlil/
  • H Short, Quasiconvexity and a theorem of Howson's, from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (É Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publishing, River Edge, NJ (1991) 168–176
  • P Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157–187
  • E R van Kampen, On some lemmas in the theory of groups, Amer. J. Math. 55 (1933) 268–273
  • C M Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. Soc. 30 (1971) 22–26
  • J Wilson, A CAT(0) group with uncountably many distinct boundaries, preprint
  • D T Wise, Non-positively curved squared complexes, aperiodic tilings, and non-residually finite groups, PhD thesis, Princeton Univ. (1996)
  • D T Wise, Subgroup separability of the figure 8 knot group (1998), preprint available at http://www.math.mcgill.ca/wise/papers.html