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

We prove the existence of surface subgroups within any cocompact lattice $\Gamma$ in $\mathrm{SO}(2n,1)$ for $n\geq2$. This result addresses the cases missing from the work of Hamenst\"adt in 2015, who constructed surface subgroups in cocompact lattices for all other rank-one semisimple Lie groups of non-compact type.