Reflections, bendings, and pentagons

Details Reflections, bendings, and pentagons Reflections, bendings, and pentagons

2012-01-07

arxiv.org/abs/1201.1582v2

Mathematics

Metric Geometry

18 pages, 8 references. Full abstract inserted in metadata

Scientific paper

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 q_1,q_2 obtained from p_1,p_2 by moving p_1,p_2 along the geodesic generated by p_1,p_2 and keeping the (dis)tance between p_1,p_2 satisfy the bending relation R(q_2)R(q_1)=R(p_2)R(p_1). We show that a generic isometry F\in SU(2,1) is a product of 3 reflections, F=R(p_3)R(p_2)R(p_1), and describe all such decompositions: two decompositions are connected by finitely many bendings involving p_1,p_2/p_2,p_3 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 r_1,...,r_n with the defining relations r_n...r_1=1, r_j^2=1. The theorem mentioned above is essential to the study of the Teichmuller space TH_n. We describe all nontrivial representations of H_5, called pentagons, and conjecture that they are faithful and discrete.

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