Complex hyperbolic free groups with many parabolic elements

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.