# Complex hyperbolic free groups with many parabolic elements.

* Corresponding author
Abstract : We consider in this work representations of the of the fundamental group of the 3-punctured sphere in ${\rm PU}(2,1)$ such that the boundary loops are mapped to ${\rm PU}(2,1)$. We provide a system of coordinates on the corresponding representation variety, and analyse more specifically those representations corresponding to subgroups of $(3,3,\infty)$-groups. In particular we prove that it is possible to construct representations of the free group of rank two $\la a,b\ra$ in ${\rm PU}(2,1)$ for which $a$, $b$, $ab$, $ab^{-1}$, $ab^2$, $a^2b$ and $[a,b]$ all are mapped to parabolics.
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Cited literature [17 references]

https://hal.archives-ouvertes.fr/hal-00918321
Contributor : Pierre Will Connect in order to contact the contributor
Submitted on : Friday, December 13, 2013 - 12:13:15 PM
Last modification on : Tuesday, May 11, 2021 - 11:36:03 AM
Long-term archiving on: : Friday, March 14, 2014 - 4:35:36 AM

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### Identifiers

• HAL Id : hal-00918321, version 1
• ARXIV : 1312.3795

### Citation

John R. Parker, Pierre Will. Complex hyperbolic free groups with many parabolic elements.. Geometry, groups and dynamics. , 639, AMS, pp.327-348, 2015, Contemporary Mathematics, 978-0-8218-9882-6. ⟨hal-00918321⟩

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