A quadratic bound on the number of boundary slopes of essential surfaces with bounded genus

TL;DR

This paper establishes a quadratic upper bound on the number of boundary slopes of immersed essential surfaces with bounded genus in orientable 3-manifolds with torus boundary, extending previous hyperbolic case results.

Contribution

It provides a general quadratic bound for all orientable 3-manifolds with a torus boundary, generalizing earlier hyperbolic-specific results.

Findings

Number of boundary slopes is bounded quadratically by genus

Bound applies to immersed essential surfaces in 3-manifolds

Extends hyperbolic case results to general orientable 3-manifolds

Abstract

Let $M$ be an orientable 3-manifold with $\partial M$ a single torus. We show that the number of boundary slopes of immersed essential surfaces with genus at most $g$ is bounded by a quadratic function of $g$. In the hyperbolic case, this was proved earlier by Hass, Rubinstein and Wang.