Groups, Geometry, and Dynamics


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Volume 8, Issue 2, 2014, pp. 355–374
DOI: 10.4171/GGD/229

Published online: 2014-07-08

A note on trace fields of complex hyperbolic groups

Heleno Cunha[1] and Nikolay Gusevskii[2]

(1) Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
(2) Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

We show that if $\Gamma$ is an irreducible subgroup of SU(2,1), then $\Gamma$ contains a loxodromic element $A$. If $A$ has eigenvalues $\lambda_1 = \lambda e^{i\varphi}$, $\lambda_2 = e^{-2i\varphi}$, $\lambda_3 = \lambda^{-1}e^{i\varphi}$, we prove that $\Gamma$ is conjugate in SU(2,1) to a subgroup of SU$(2,1,\mathbb{Q}(\Gamma,\lambda))$, where $\mathbb{Q}(\Gamma, \lambda)$ is the field generated by the trace field $\mathbb{Q}(\Gamma)$ of $\Gamma$ and $\lambda$. It follows from this that if $\Gamma$ is an irreducible subgroup of SU(2,1) such that the trace field $\mathbb{Q}(\Gamma)$ is real, then $\Gamma$ is conjugate in SU(2,1) to a subgroup of SO(2,1). As a geometric application of the above, we get that if $G$ is an irreducible discrete subgroup of PU(2,1), then $G$ is an $\mathbb R$-Fuchsian subgroup of PU(2,1) if and only if the invariant trace field $k(G)$ of $G$ is real.

Keywords: Complex hyperbolic groups, trace fields

Cunha Heleno, Gusevskii Nikolay: A note on trace fields of complex hyperbolic groups. Groups Geom. Dyn. 8 (2014), 355-374. doi: 10.4171/GGD/229