A characterization of quaternionic Kleinian groups in dimension 2 with complex trace fields

TL;DR

This paper characterizes certain quaternionic Kleinian groups in dimension 2, showing that if their elements have complex trace fields, they are conjugate to subgroups of a unitary group, revealing structural constraints.

Contribution

It establishes a criterion based on trace fields that determines when quaternionic Kleinian groups are conjugate to complex unitary groups in dimension 2.

Findings

Groups with complex trace fields are conjugate to subgroups of U(2,1)

Provides a characterization of quaternionic Kleinian groups in dimension 2

Links trace field properties to group conjugacy classes

Abstract

Let $G$ be a non-elementary discrete subgroup of $\mathrm{Sp}(2,1)$. We show that if the sum of diagonal entries of each element of $G$ is a complex number, then $G$ is conjugate to a subgroup of $\mathrm{U}(2,1)$.