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

TL;DR

This paper investigates the properties of trace fields of subgroups within SU(n,1), revealing conditions under which the trace field aligns with the field generated by matrix coefficients and linking it to geometric invariants.

Contribution

It establishes conditions where the trace field equals the coefficient field for subgroups of SU(n,1), especially in geometric and arithmetic contexts.

Findings

Trace field equals coefficient field under certain hypotheses.

Relation between trace field and geometric invariants for 3-manifold groups.

For arithmetic groups, trace field and coefficient field coincide.

Abstract

In this note, we study the field generated by the traces of subgroups of SU(n,1). Under some hypotheses, the trace field of a group $\Gamma \subset$ SU(2,1) is equal to the field generated by the coefficients of the matrices in $\Gamma$. If the group is the image of a representation of the fundamental group of a triangulated 3-manifold, we can relate the trace field to a geometric invariant. For an arithmetic group of the first type in SU(n,1), up to conjugacy, the trace field and the field of the coefficients are the same.