This is a joint work with J. Behrstock and C. Drutu. We show that if a group G has infinitely many pairwise non-conjugate homomorphisms into a mapping class group of a surface, then G has a finite index subgroup (the index depends on the surface only) that acts non-trivially on a real tree. If G is finitely presented, then the real tree can be replaced by a simplicial tree.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/249