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

Abstract

Let $d$ be a positive square-free integer $\equiv 3(\mathrm{mod}4)$ such that there is no invariant of the ideal class group $\mathbb{Q}[\sqrt{-d}]$ 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 $X$.

10 pages