Margulis numbers for Haken manifolds

TL;DR

This paper establishes specific Margulis numbers for hyperbolic Haken 3-manifolds, showing that 0.286 or 0.292 serve as Margulis constants depending on the manifold's properties, advancing understanding of their geometric structure.

Contribution

The paper proves explicit Margulis numbers for broad classes of hyperbolic Haken 3-manifolds, including cases with non-zero Betti number or semi-fibers, extending prior results.

Findings

0.286 is a Margulis number for all hyperbolic Haken 3-manifolds.

0.292 is a Margulis number if the manifold has non-zero first Betti number or contains a semi-fiber.

The results provide explicit bounds for Margulis numbers in these classes.

Abstract

For every closed hyperbolic Haken 3-manifold and, more generally, for any hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.