On the growth of the number of totally geodesic surfaces in some hyperbolic $3$-manifolds

TL;DR

This paper establishes an asymptotic formula for counting immersed totally geodesic surfaces of bounded area in certain hyperbolic 3-manifolds associated with specific imaginary quadratic fields.

Contribution

It provides the first asymptotic count of such surfaces in hyperbolic 3-manifolds linked to imaginary quadratic fields with particular class group properties.

Findings

Asymptotic formula for the number of geodesic surfaces

Conditions on the quadratic field for the count

Growth rate of the number of surfaces with area

Abstract

Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed totally geodesic surfaces in $\Gamma_{-d}\backslash \mathbb{H}^3$ having area less than $X$.