Rigidity of complex bounded domains with finite-volume quotients

We prove that any bounded convex domain in ${\mathbb{C}}^n$ admitting a finite-volume quotient is biholomorphic to a bounded symmetric domain. We first show that such a convex domain always has a nontrivial one-parameter family of automorphisms; the proof then relies on our solution to Kazhdan’s conjecture for contractible HHR/USq domains with finite-volume quotients. More precisely, we prove that any contractible bounded domain that has the uniform squeezing property and admits a finite-volume quotient is biholomorphic to a product of a bounded symmetric domain and a contractible complex manifold with discrete automorphism group. The quotient groups are discrete and may have finite stabilizers; volume is measured with respect to the complete K"ahler–Einstein metric. As a further application, we also show that the automorphism groups of Bers–Griffiths domains in ${\mathbb{C}}^2$, which are not biholomorphic to the bidisk, are discrete.