Reflections, bendings, and pentagons

2012/01/07 by Anan'in, Sasha
#30F60 (Primary) 51M10 #57S30 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1201.1582

Abstract

We study relations between reflections in (positive or negative) points in the complex hyperbolic plane. It is easy to see that the reflections in the points q₁,q₂ obtained from p₁,p₂ by moving p₁,p₂ along the geodesic generated by p₁,p₂ and keeping the (dis)tance between p₁,p₂ satisfy the bending relation R(q₂)R(q₁)=R(p₂)R(p₁). We show that a generic isometry F∈ SU(2,1) is a product of 3 reflections, F=R(p₃)R(p₂)R(p₁), and describe all such decompositions: two decompositions are connected by finitely many bendings involving p₁,p₂/p₂,p₃ and geometrically equal decompositions differ by an isometry centralizing F. Any relation between reflections gives rise to a representation H_n->PU(2,1) of the hyperelliptic group H_n generated by r₁,...,r_n with the defining relations r_n...r₁=1, r_j²=1. The theorem mentioned above is essential to the study of the Teichmuller space TH_n. We describe all nontrivial representations of H₅, called pentagons, and conjecture that they are faithful and discrete.

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