Teichmüller modular groups and arithmetical groups

In this paper we give a negative answer to a question of W. Harvey about the arithmeticity of Teichmüller modular forms and we prove a number of results about the nonarithmeticity of subnormal subgroups of modular groups. In the Appendix we announce and discuss a theorem according to which the appropriately defined rank of Teichmüller modular groups is equal to 1.