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Negative curvature and combinatorial group theory

Published online by Cambridge University Press:  09 April 2009

Joseph A. Wolf
Joseph A. Wolf
Affiliation:
University of California Berkeley, California 94720, U.S.A.
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Combinatorial group theory has roots in Poincaré's work on the topology of manifolds, which in turn was based on problems in differential equations and analytic number theory. Thus the Fuchsian groups, which are the fundamental (first homotopy) groups of oriented negatively curved compact surfaces, served as important models in their day. In the last few years there have been advances in the understanding of the structure of fundamental groups of negatively curved manifolds, some of them based on examples from analytic number theory. Here I describe one of these developments and pose a few difficult combinatorial questions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 1 , August 1973 , pp. 14 - 15
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Gromoll, D. and Wolf, J. A., ‘Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature’, Bull. Amer. Math. Soc. 77 (1971), 545–552.Google Scholar
[2]Milnor, J., ‘Growth of finitely generated solvable groups’, J. Differential Geometry 2 (1968), 447–449.Google Scholar
[3]Wolf, J. A., ‘Growth of finitely generated solvable groups and curvature of Riemannian manifolds’, J. Differential Geometry 2 (1968), 421–446.Google Scholar