Abstract
Given two 3-dimensional handlebodies whose boundaries are identified with a surface S of genus g > 1 and with different orientations, we consider the sequence of manifolds M n obtained by gluing the handlebodies via the iteration f n of a “generic” pseudo-Anosov homeomorphism f of S. Using the deformation theory of hyperbolic structures on open hyperbolic 3-manifolds and for n sufficiently large, we construct a negatively curved metric on M n where the sectional curvatures are pinched in a given small interval centered at –1. The construction is concrete enough to allow us describe the geometric limits of these manifolds as n tends to infinity and the metrics get closer to being hyperbolic. Such a description allows us to prove various topological and group theoretical properties of M n , for n sufficiently large, which would not be available knowing the mere existence of a negatively curved or even hyperbolic metric on M n .
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The first author was partially supported by an award of the Clay Mathematics Institute and by National Science Foundation Grant 0604111. The second author was partially supported by National Science Foundation Grant 0706878 and Alfred P. Sloan Foundation.
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Namazi, H., Souto, J. Heegaard Splittings and Pseudo-Anosov Maps. Geom. Funct. Anal. 19, 1195–1228 (2009). https://doi.org/10.1007/s00039-009-0025-3
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DOI: https://doi.org/10.1007/s00039-009-0025-3