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 .
Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.
We show that if M is a closed three manifold with a Heegaard splitting with sufficiently big Heegaard distance then the subgroup of the mapping class group of the Heegaard surface, whose elements extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of M is finite.
We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the folding/unfolding path. It follows that non-unique ergodicity leads to distortion when projecting to the complex of free factors, and we give two applications of this fact. First, we show that if a subgroup H of Out(FN ) quasi-isometrically embeds into the complex of free factors via the orbit map, then the limit set of H in the boundary of Outer space consists of trees that are uniquely ergodic and have uniquely ergodic dual lamination. Second, we describe the Poisson boundary for random walks coming from distributions with finite first moment with respect to the word metric on Out(FN ): almost every sample path converges to a tree that is uniquely ergodic and that has a uniquely ergodic dual lamination, and the corresponding hitting measure on the boundary of Outer space is the Poisson boundary. This improves a recent result of Horbez. We also obtain sublinear tracking of sample paths with Lipschitz geodesic rays.
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