Complex and Quaternionic hyperbolic Kleinian groups with real trace fields

TL;DR

This paper investigates discrete subgroups of SU(n,1) and Sp(n,1) with real trace fields, showing they preserve certain geometric structures and relate to subgroups of SO(n,1), extending Maskit's theorem to rank 1 Lie groups.

Contribution

It establishes a connection between real trace fields and geometric preservation properties for hyperbolic Kleinian groups, generalizing Maskit's theorem to complex and quaternionic settings.

Findings

Groups with real trace fields preserve totally geodesic submanifolds.

Irreducible groups with real trace fields are conjugate to subgroups of SO(n,1).

The results extend classical theorems to complex and quaternionic hyperbolic groups.

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 $1$.