Two Generator groups acting on the complex hyperbolic plane

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.Comment: 49 pages, 10 figures. It will appear as a chapter of Volume VI of the Handbook of Teichmuller theor

On SL(3,C\mathbb C)-representations of the Whitehead link group

We describe a family of representations in SL(3,$\mathbb C$) of the fundamental group $\pi$ of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,$\mathbb C$) and can be seen as factorising through a quotient of $\pi$ defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,$\mathbb C$)-character variety of $\pi$.Comment: 20 pages, 3 figures, 4 tables, and a companion Sage notebook (see the references) v2: A few corrections and improvement

Involution and commutator length for complex hyperbolic isometries

We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \geqslant 3$ PU($n$,1) has involution length at most 8.Comment: 32 pages, 22 figure

Complex hyperbolic free groups with many parabolic elements

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.Comment: 21 pages, 11 figure

Real reflections, commutators and cross-ratios in complex hyperbolic space

26 pagesInternational audienceWe provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8

Complex hyperbolic free groups with many parabolic elements

We consider in this work representations of the of the fundamental group of the 3-punctured sphere in PU(2,1) such that the boundary loops are mapped to 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,∞) -groups. In particular we prove that it is possible to construct representations of the free group of rank two \la a,b\ra in PU(2,1) for which a , b , ab , ab −1 , ab 2 , a 2 b and [a,b] all are mapped to parabolics