Reflections, bendings, and pentagons

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 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.