Reflections, bendings, and pentagons

TL;DR

This paper explores the relations between reflections in the complex hyperbolic plane, characterizes decompositions of isometries into reflections, and studies representations of hyperelliptic groups related to pentagons, with implications for Teichmüller space.

Contribution

It provides a detailed description of isometry decompositions into reflections and classifies all pentagon representations of the hyperelliptic group, proposing their faithfulness and discreteness.

Findings

Generic isometries can be expressed as products of three reflections.

Decompositions are connected by finitely many bendings and related by centralizing isometries.

All nontrivial pentagon representations of H_5 are described and conjectured to be faithful and discrete.

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.