Lattices in $PSL_2(\mathbb{R})$

Question

Can a lattice in $PSL_2(\mathbb{R})$ have a normal abelian subgroup? It looks to me that it doesn't, but where can I read a proof?

Answer 1

A somewhat different argument from that of Yves comes from Delzant's theorem SOUS-GROUPES DISTINGUÉS ET QUOTIENTS DES GROUPES HYPERBOLIQUES, which states that the normal closure of a pair of noncommuting hyperbolic elements contains a free group. Showing that any normal subgroup contains hyperbolic elements is not hard.

Answer 2

In any semisimple connected Lie group with trivial center, any lattice is Zariski-dense. Hence any normal abelian subgroup N should have its Zariski closure normal in the whole group, which forces $N=1$.