Reflections, bendings, and pentagons
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...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
07.01.2012
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Subjects | |
Online Access | Get full text |
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Summary: | 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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DOI: | 10.48550/arxiv.1201.1582 |