Complex and Quaternionic hyperbolic Kleinian groups with real trace fields

Abstract

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb{R}$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $\Gamma$ is irreducible, $\Gamma$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.

24 pages, some changes in the introduction and theorems

References in corpus (1)

• Trace Fields of Subgroups of SU(n,1))