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 $\mathrm{PU}(2,1)$ such that the boundary loops are mapped to $\mathrm{PU}(2,1)$. 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 $\mathrm{PU}(2,1)$ for which $a$, $b$, $ab$, $a{b}^{-1}$, $a{b}^2$, $a^{2}b$ and $[a,b]$ all are mapped to parabolics.

21 pages, 11 figures