A note on complex hyperbolic lattices and strict hyperbolization

We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney–Davis and Davis–Januszkiewicz–Weinberger. Viewing a non-uniform lattice $\mathrm{\Gamma }$ in $\text{PU}(n,1)$ as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that when $n\ge 2$, $\mathrm{\Gamma }$ is not isomorphic to any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont–Ruffoni. We also prove that a uniform lattice in $\text{PU}(n,1)$ is not the fundamental group of a Charney-Davis strict hyperbolization when $n\ge 2$, assuming the initial complex satisfies some mild conditions.