Quaternionic hyperbolic Kleinian groups with commutative trace skew-fields

TL;DR

This paper proves that non-elementary discrete subgroups of quaternionic hyperbolic space with commutative trace skew-fields stabilize complex hyperbolic subspaces, revealing structural constraints of such groups.

Contribution

It establishes a link between the commutativity of the trace skew-field and the geometric stabilization properties of quaternionic hyperbolic Kleinian groups.

Findings

If the trace skew-field is commutative, then the group stabilizes a complex hyperbolic subspace.

The result characterizes the structure of certain quaternionic hyperbolic Kleinian groups.

Provides conditions under which these groups preserve complex hyperbolic geometry.

Abstract

Let $\Gamma$ be a nonelementary discrete subgroup of $\mathrm{Sp}(n,1)$. We show that if the trace skew-field of $\Gamma$ is commutative, then $\Gamma$ stabilizes a copy of complex hyperbolic subspace of quaternionic hyperbolic $n$-space.