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# Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$

Published:2009-12-01
Printed: Dec 2009
• Pierre Will
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## Abstract

In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$.
 MSC Classifications: 14L24 - Geometric invariant theory [See also 13A50] 22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 32M15 - Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15] 51M10 - Hyperbolic and elliptic geometries (general) and generalizations