Surface Subgroups for Cocompact Lattices of Isometries of $H^{2n}$

We prove the existence of surface subgroups within any cocompact lattice $\mathrm{\Gamma }$ in $\mathrm{S}\mathrm{O}(2n,1)$ for $n\ge 2$. This result addresses the cases missing from the work of Hamenstädt in 2015, who constructed surface subgroups in cocompact lattices for all other rank-one semisimple Lie groups of non-compact type.