A note on the integrality of volumes of representations

Let $\mathrm{\Gamma }$ be a torsion-free, non-uniform lattice in $\mathrm{S}\mathrm{O}(2n,1)$. We present an elementary, combinatorial-geometrical proof of a theorem of Bucher, Burger, and Iozzi which states that the volume of a representation $\rho :\mathrm{\Gamma }\to \mathrm{S}\mathrm{O}(2n,1)$, properly normalized, is an integer if $n$ is greater than or equal to $2$.