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

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 $X$.