Splitting fields of complex hyperbolic groups: matrix extraction, Hermitian descent, and spectral fields

Source abstract

Let $Γ<\mathrm{SU}(n,1)$, $n\ge2$, be Zariski-dense and contain a regular loxodromic element $A$. The splitting-field problem asks how far traces and spectra determine a field of definition for $Γ$. We show first that traces together with the spectrum of $A$ always give a linear realization over $K=\mathbb{Q}(\textrm{tr}Γ,E(A))$. We also prove that every complex-irreducible subgroup of $\mathrm{SU}(n,1)$ with real trace field is conjugate into $\mathrm{SO}(n,1)$; in particular, no discreteness hypothesis is required for this real-trace conclusion. In higher dimension, however, linear descent does not by itself give the standard unitary form: a Hermitian similarity obstruction remains, described over imaginary quadratic fields by Landherr's local invariants. In $\mathrm{SU}(3,1)$ we give a Zariski-dense example for which the one-spectrum field is $\mathbb{Q}(i)$ but does not yield the standard unitary form. This leads naturally to the total spectral field. We prove that the obstruction disappears for arithmetic groups and under suitable local openness hypotheses, and we give an even-degree local criterion. In dimension four we analyze the discrete branch through explicit Schottky groups and rank-one local geometry. There the full Bruhat cocycle recovers anisotropic Levi holonomy and forces a non-split rotational word. The remaining quasi-split rank-two case is not reached by this method, so the general $\mathrm{SU}(3,1)$ problem remains open.