Sous-groupes discrets de PU(2,1) engendrés par n réflexions complexes et Déformation

Abstract

Let PU (2, 1) be the group of holomorphic isometries in the hyperbolic complex plane H 2 C and let G n be a sub-group of PU (2, 1) which is generated by n complex reflections with respect to complex lines in H 2 C . Under certain conditions, we prove that G n is discrete. We construct representations ρ of the fundamental group g of the compact surface g of genus g, into PU (2, 1), we prove they are discrete, faithful and we compute the dimension their deformation space.

Two-generator groups acting on the complex hyperbolic plane

“…These are the only two cases where an isometry can have more than one fixed point in H 2 C . We see thus that the contribution of ρ(c j ) to the right hand side product of (35) does not depent on the chosen fixed point.…”

“…Explicit spherical CR structure on circle bundles over hyperbolic surfaces are relatively easy to produce by considering discrete and faithful representations of surface groups in PU (2,1). Many examples can be found in the litterature (among these [1,2,39,35]). In [101], Schwartz has given an example of a spherical CR structure on a closed hyperbolic 3 manifold.…”

Bending Fuschsian representations of fundamental groups of cusped surfaces in $\mathrm{PU}(2,1)$

“…The question of discreteness of representations of surface groups in PU(n,1) is still far from being given a complete answer. It is known for instance that contrary to the case of PSL(2,R), discrete and faithful representations are not contained in specific components of Rep Σ,PU (2,1) , as shown for instance in [19] or [1,2,15]. This work is concerned with representations of fundamental groups of cusped surfaces.…”