Polynomially many surfaces of fixed Euler characteristic in a hyperbolic 3-manifold

TL;DR

This paper establishes a polynomial upper bound on the number of certain essential surfaces in hyperbolic 3-manifolds, linking the count to the manifold's volume and the surfaces' Euler characteristic.

Contribution

It provides the first polynomial bound on the number of such surfaces in hyperbolic 3-manifolds, depending linearly on the Euler characteristic.

Findings

Number of surfaces is polynomial in the volume of the manifold.

Degree of polynomial depends linearly on the absolute value of Euler characteristic.

Bound applies to both closed and cusped hyperbolic 3-manifolds.

Abstract

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This bound is a polynomial function of the volume of $M$, with degree that depends linearly on $|\chi|$.