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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
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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

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