Length and stable length

Details Length and stable length Length and stable length

2006-05-15

arxiv.org/abs/math/0605354v4

Geom. Func. Anal. 18 (2008), no. 1, 50-76

Mathematics

Geometric Topology

22 pages, 2 figures; version 4: incorporates referee's suggestions

Scientific paper

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive e there is a positive d depending only on n and on e such that an element of pi_1(M) with stable commutator length less than d is represented by a geodesic with length less than e. Moreover, for any such M, the first accumulation point for stable commutator length on conjugacy classes is at least 1/12. Conversely, "most" short geodesics in hyperbolic 3-manifolds have arbitrarily small stable commutator length. Thus stable commutator length is typically good at detecting the thick-thin decomposition of M, and 1/12 can be thought of as a kind of homological Margulis constant.

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