Complex hyperbolic free groups with many parabolic elements.

John R. Parker (1) , Pierre Will (2)

1 Durham University - Department of Mathematical Sciences
2 IF - Institut Fourier

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.

Mots clés: Complex hyperbolic geometry; discrete subgroups of Lie groups

Domaines: Topologie géométrique [math.GT]

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⟩