A classical theorem of Wang says that for every positive real number $v$ the number $N(v)$ of distinct (up to an isometry) hyperbolic $n$-manifolds with volume bounded by $v$, provided $n\geq 4$, is finite. The recent results by Burger-Gelander-Lubotzky-Mozes, Belolipetsky-Gelander-Lubotzky-Shalev, Gelander-Levit show that there are super-exponentially many non-isometric hyperbolic manifolds with respect to volume as a ""complexity measure"", i.e. $a_1 v^{b_1 v} \leq N(v) \leq a_2 v^{b_2}v$ (if $v$ is large enough), and an analogous statement holds for the number of arithmetic hyperbolic $n$-manifolds, and even the number of their commensurability classes.

We shall continue in this direction and investigate some particular families of $4$-dimensional hyperbolic manifolds with more specific properties (still not entirely accessible in arbitrary dimensions), such as having a given number of cusps, a given symmetry group, and other natural geometric or topological restrictions. We shall show that in all cases the number of such manifolds grows super-exponentially with respect to volume.

This talk is based on the joint papers and work in progress with Bruno Martelli (University of Pisa), Leone Slavich (University of Pisa), Steven Tschantz (Vanderbilt University), Alan Reid (Rice University) and Stefano Riolo (University of Neuchâtel).