Three Generalizations Regarding Limit Sets for Complex Kleinian Groups

Abstract

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.