On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds

TL;DR

This paper proves that hyperbolic 3-manifolds with totally geodesic boundary have finitely many boundary slopes for essential immersed surfaces of fixed genus, with bounds depending on volume and boundary genus.

Contribution

It establishes finiteness and uniform bounds on boundary slopes for essential immersed surfaces in hyperbolic 3-manifolds with totally geodesic boundary.

Findings

Finiteness of boundary slopes for fixed genus surfaces.

Uniform bounds when volume or boundary genus is bounded.

Bounded boundary area and geodesic lengths under volume constraints.

Abstract

For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of $\partial M$ or the volume of $M$ is bounded above. When the volume is bounded above, then area of $\partial M$ is bounded above and the length of closed geodesic on $\partial M$ is bounded below.