Joseph Wells

Complex hyperbolic hybrids

The celebrated works of Margulis, Gromov--Schoen, and Corlette collectively imply that lattices in simple Lie groups are always arithmetic, with the lone exceptions of the isometry groups for the real and complex hyperbolic spaces. Like Nessy and Bigfoot, non-arithmetic lattices are fairly mysterious and much work has gone into finding them (we have more than blurry photographs to prove such lattices exist). In the 1980's, Gromov and Piatetski-Shapiro produced a geometric technique called "hybridization" in which one constructs non-arithmetic lattices in the real hyperbolic isometries (in any dimension) starting from two non-commensurable arithmetic lattices (of the same dimension). One can ask whether there exists an analogous technique for the complex hyperbolic isometries, and in this talk I'll present one potential candidate and some recent results obtained from it.