Heegaard splittings and pseudo-Anosov maps. (English) Zbl 1210.57019

Namazi, Hossein; Souto, Juan

Geom. Funct. Anal. 19, No. 4, 1195-1228 (2009).

Heegaard splittings of \(3\)-manifolds have been studied mostly using topological or combinatorial methods. This paper is one of the first attempts to use the techniques of hyperbolic geometry for studying these objects.
The authors consider \(3\)-manifolds \(M_n\) obtained by attaching two handlebodies (of genus \(g\) greater than \(1\)) by iterations of pseudo-Anosov maps \(f^n\) along their boundaries. It is assumed that the stable lamination of \(f\) is not contained in the closure of the set of meridians in the projective lamination space. They show that for any pseudo-Anosov map \(f\), if we take sufficiently large \(n\), then \(M_n\) admits a negatively curved metric whose sectional curvature is pinched within a small interval containing \(-1\). They also show that a geometric limit of \(M_n\) is homeomorphic to either a handlebody of genus \(g\) or a product open-interval bundle over the closed surface of genus \(g\).
There are several applications of these theorems. One of them claims that the rank of \(\pi_1(M_n)\) is equal to \(g\) for sufficiently large \(n\), and every minimal generating set of \(\pi_1(M_n)\) is Nielsen equivalent to one of the two standard generating sets.
The proofs of the main theorems use results and techniques which were recently developed in the field of Kleinian groups and hyperbolic 3-manifolds.

Reviewer: Ken’ichi Ohshika (Osaka)

MSC:

57M50	General geometric structures on low-dimensional manifolds
57M07	Topological methods in group theory
57N10	Topology of general \(3\)-manifolds (MSC2010)
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)

Keywords:

Heegaard splitting; pseudo-Anosov map; hyperbolic 3-manifold

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