Hyperbolization of infinite-type 3-manifolds

TL;DR

This paper investigates conditions under which certain infinite-type 3-manifolds can be endowed with complete hyperbolic metrics, focusing on manifolds built from hyperbolizable pieces with controlled boundary complexity.

Contribution

It provides necessary and sufficient topological conditions for infinite-type 3-manifolds in a specific class to admit complete hyperbolic metrics.

Findings

Characterization of manifolds in $\\mathcal M^B$ that admit hyperbolic metrics

Topological criteria for hyperbolization of infinite-type 3-manifolds

Extension of hyperbolization results to manifolds with boundary components of bounded genus

Abstract

We study the class $\mathcal M^B$ of 3-manifolds $M$ that have a compact exhaustion $M=\cup_{i\in\mathbb N} M_i$ satisfying: each $M_i$ is hyperbolizable with incompressible boundary and each component of $\partial M_i$ has genus at most $g= g(M)$. For manifolds in $\mathcal M^{B}$ we give necessary and sufficient topological conditions that guarantee the existence of a complete hyperbolic metric.