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

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 $|\chi|$.