Hyperbolization of infinite-type 3-manifolds

Abstract

We study the class M B of 3-manifolds M that have a compact exhaustion M = ∪ i∈N M i satisfying: each M i is hyperbolizable with incompressible boundary and each component of ∂M i has genus at most g = g(M ). For manifolds in M B we give necessary and sufficient topological conditions that guarantee the existence of a complete hyperbolic metric.

Effective contraction of Skinning maps

“…Indeed, consider all manifolds M ∈ M B such that for all i ∈ N every component U i := M i \ M i−1 is acylindrical. By the main results of [6] this guarantees that M is in fact hyperbolic, which is in general not the case, see [5,7], or [8,9] for other examples of infinite-type hyperbolic 3-manifolds. We can thus think of a (hyperbolic) metric g on M as a gluing of (hyperbolic) metrics g i on the U i 's and so it makes sense to investigate the glueing of pairs U i , U i+1 via skinning maps.…”

“…An application to infinite-type 3-manifolds. In [6] the first author studied the class M B of infinite-type 3-manifolds M admitting an exhaustion M = ∪ i M i by hyperbolizable 3-manifolds M i with incompressible boundary and with uniformly bounded genus.…”

Untitled

“…Indeed, consider all manifolds M ∈ M B such that for all i ∈ N every component U i := M i \ M i−1 is acylindrical. By the main results of [6] this guarantees that M is in fact hyperbolic, which is in general not the case, see [5,7], or [8,9] for other examples of infinite-type hyperbolic 3-manifolds. We can thus think of a (hyperbolic) metric g on M as a gluing of (hyperbolic) metrics g i on the U i 's and so it makes sense to investigate the glueing of pairs U i , U i+1 via skinning maps.…”

“…An application to infinite-type 3-manifolds. In [6] the first author studied the class M B of infinite-type 3-manifolds M admitting an exhaustion M = ∪ i M i by hyperbolizable 3-manifolds M i with incompressible boundary and with uniformly bounded genus.…”