Arithmetic properties of discrete subgroups

That the factor space of a semisimple Lie group by an arithmetic subgroup has finite volume with respect to Haar measure is well known. In this paper we study results related to the converse of this theorem. In particular, under some rather weak assumptions on a semisimple Lie group $G$ we prove that every discrete subgroup of $G$ with a non-compact factor space of finite volume that satisfies some natural irreducibility conditions, is an arithmetic subgroup of $G$. In this paper we also study various results from the theory of algebraic groups and their arithmetic and discrete subgroups. In the proof of one theorem we use a construction from representation theory that is of independent interest. At the end we state some unsolved problems in the theory of discrete subgroups.