A note on trace fields of complex hyperbolic groups

2013/03/07 by Heleno Cunha, Cunha, Heleno, Nikolay Gusevskii +1
Mathematics · #20H10 #22E40 #32C16 #32G07 #32H20 #57S30 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:20H10 #msc:22E40 #msc:32C16 #msc:32G07 #msc:32H20 #msc:57S30

paper · pdf · doi:10.48550/arxiv.1303.1701

To appear in Groups, Geometry, and Dynamics

arxiv created 2013/03/07 · arxiv updated 2013/03/08

Abstract

We show that if Γ is an irreducible subgroup of \rm SU(2,1), then Γ contains a loxodromic element A. If A has eigenvalues λ₁ = λe^iφ, λ₂ = e^-2iφ, λ₃ = λ^-1e^iφ, we prove that Γ is conjugate in \rm SU(2,1) to a subgroup of \rm SU(2,1,ℚ(Γ,λ)), where ℚ(Γ, λ) is the field generated by the trace field ℚ(Γ) of Γ and λ. It follows from this that if Γ is an irreducible subgroup of \rm SU(2,1) such that the trace field ℚ(Γ) is real, then Γ is conjugate in \rm SU(2,1) to a subgroup of \rm SO(2,1). As a geometric application of the above, we get that if G is an irreducible discrete subgroup of \rm PU(2,1), then G is an ℝ-Fuchsian subgroup of \rm PU(2,1) if and only if the invariant trace field k(G) of G is real.

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