Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T18:54:05.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundaries of nonpositively curved groups of the form G × ℤn

Published online by Cambridge University Press:  18 May 2009

Philip L. Bowers  and
Kim Ruane
Philip L. Bowers
Affiliation:
Department of Mathematics 3027, Florida State University, Tallahassee, Florida, 32306-3027, USA
Kim Ruane
Affiliation:
Department of Mathematics 3027, Florida State University, Tallahassee, Florida, 32306-3027, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The introduction of curvature considerations in the past decade into Combinatorial Group Theory has had a profound effect on the study of infinite discrete groups. In particular, the theory of negatively curved groups has enjoyed significant and extensive development since Cannon's seminal study of cocompact hyperbolic groups in the early eighties [7]. Unarguably the greatest influence on the direction of this development has been Gromov's tour de force, his foundational essay in [12] entitled Hyperbolic Groups. Therein Gromov hints at the prospect of developing a corresponding theory of “non-positively curved groups” in his non-definition (Gromov's terminology) of a semihyperbolic group as a group that “looks as if it admits a discrete co-compact isometric action on a space of nonpositive curvature”; [12, p. 81]. Such a development is now occurring and is closely related to the other notable outgrowth of the theory of negatively curved groups, that of automatic groups [10]; we mention here the works [3] and [6] as developments of a theory of nonpositively curved groups along with Chapter 6 of Gromov's more recent treatise [13]. A natural question that serves both to guide and organize the developing theory is: to what extent is the well-developed theory of negatively curved groups reflected in and subsumed under the developing theory of nonpositively curved groups? Our overall interest is in one aspect of this question—namely, as the question relates to the boundaries of groups and spaces: can one define the boundary of a nonpositively curved group intrinsically in a way that generalizes that of negatively curved groups and retains some of their essential features?

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 38 , Issue 2 , May 1996 , pp. 177 - 189
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

REFERENCES

1.Alexandrov, A. D., Berestovskii, V. N., and Nikolaev, I. G., Generalized Riemannian spaces, Russ. Math. Surveys 41:3 (1986), 1–54.CrossRefGoogle Scholar
2.Alonso, J., Brady, T., Cooper, D., Delzant, T., Lustig, M., Mahilik, M., Shapiro, M., Short, H., and Other, A. N., Notes on word hyperbolic groups, in Group Theory from a Geometrical Viewpoint (Ghys, E., Haefliger, A., Verjovsky, A., Eds.), (Word Scientific, 1991), 3–63.Google Scholar
3.Alonso, J. and Bridson, M., Semihyperbolic groups, Proc. London Math. Soc. (3) 70 (1995), 56–114.CrossRefGoogle Scholar
4.Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of nonpositive curvature, (Birkhauser, 1985).CrossRefGoogle Scholar
5.Bowers, P. L. and Ruane, K., Fixed points in boundaries of negatively curved groups, Proc. Amer. Math. Soc., to appear.Google Scholar
6.Bridson, M. and Haefliger, A., book in preparation.Google Scholar
7.Cannon, J., The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Ded. 16 (1984), 123–148.Google Scholar
8.Cannon, J., The theory of negatively curved spaces and groups, Chapter 11 of Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Bedford, T., Keane, M. and Series, C., Eds.) (Oxford Univ. Press 1991), 315–369.Google Scholar
9.Coornaert, M. and Papadopoulos, A., Symbolic Dynamics and Hyperbolic Croups, Lecture Notes in Mathematics No. 1539 (Springer-Verlag, 1993).Google Scholar
10.Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P., Word Processing in Groups, (Jones and Bartlett 1992).CrossRefGoogle Scholar
11.Ghys, E. and de la Harpe, P., Infinite groups as geometric objects (after Gromov), Chapter 10 of Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Bedford, T., Keane, M. and Series, C., Eds.), (Oxford Univ. Press 1991) 299–314.Google Scholar
12.Gromov, M., Hyperbolic groups, in Essays in Group Theory (Gersten, S. M., Ed.), (Springer-Verlag, 1987), 75–263.CrossRefGoogle Scholar
13.Gromov, M., Asymptotic invariants of infinite groups, in Geometric Group Theory: Proceedings of the Symposium Held in Sussex, 1991, (Cambridge Univ. Press, 1993).Google Scholar
14.Mihalik, M., Semistability of Artin and Coxeter groups, preprint.Google Scholar
15.Moussong, G., Hyperbolic Coxeter groups, PhD Thesis, (Ohio State University, 1988).Google Scholar