  We study three problems related to the limit sets of discrete subgroups of
PSL(n+1,C).
  In Chapter 2, we study the dynamics of solvable discrete subgroups of
PSL(n+1,C). We prove that solvable groups are virtually triangularizable and we
provide a description of the all the possible Kulkarni limit sets of solvable
subgroups of PSL(n+1,C). Finally, we give the representations of these groups.
With this description, the full description of the dynamics of general discrete
subgroups of PSL(n+1,C) will be almost complete.
  In Chapter 3, we propose a new definition for the concept of limit set for
the action of a discrete subgroup of PSL(n+1,C), we call it the Frances limit
set. In complex dimension n=2, the Kulkarni limit set seems to be the right
notion of limit set. However, in dimension n>2, the Kulkarni limit set is
difficult to compute and it is bigger than it needs to be. This new limit set
is, in general, smaller than the Kulkarni limit set. Also, it is made up of
projective subspaces of the same dimension. The action of a discrete subgroup
of PSL(n+1,C) on the complement of its Frances limit set is proper and
discontinuous, also, this limit set is purely dimensional and unstable under
deformations.
  In Chapter 4, we propose a way to generalize Patterson-Sullivan measures to
the complex setting. We consider the Kobayashi metric on the complement of the
Kulkarni limit set of an irreducible subgroup of PSL(3,C). These domains are
the complement of arrays of complex lines in general position. We parametrize
the space of such arrays of lines and we prove that, if for some subgroup of
PSL(3,C) the entropy volume is finite, then we can construct these measures. We
also give some concrete ideas on how to guarantee that the entropy volume of
the Kobayashi metric is finite for certain groups.
