A norm on the fundamental group of non-Haken 3-manifolds

Jones, Kerry N.

doi:10.1090/S0002-9939-1994-1186989-1

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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A norm on the fundamental group of non-Haken $3$-manifolds
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by Kerry N. Jones PDF

Proc. Amer. Math. Soc. 120 (1994), 305-309 Request permission

Abstract:

A canonical (presentation-independent) conjugacy-invariant norm is constructed on the fundamental group of any $3$-manifold which is orientable, irreducible, has infinite fundamental group, and contains no incompressible surface. More generally, this norm exists on any torsion-free group whose commutator quotient is finite. This norm is then computed explicitly in an example which shows that the induced metric on the group is not quasi-isometric to any word metric.

References

John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619