Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups

In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasiisometric rigidity theorem for the mapping class group MCG(S g ) of a once punctured surface S g : if K is a finitely generated group quasi-isometric to MCG(S g ) then there is a homomorphism K → MCG(S g ) with finite kernel and finite index image. This theorem is joint with Kevin Whyte. Gromov proposed the program of classifying finitely generated groups according to their large scale geometric behavior. The goal of this paper is to combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for mapping class groups of once punctured surfaces: Theorem 1 (Mosher-Whyte). If S1 g is an oriented, once-punctured surface of genus g ≥ 2 with mapping class group MCG(S1 g), and if K is a finitely generated group quasiisometric to MCG(S1 g ), then there exists a homomorphism K → MCG(S 1 g ) with finite kernel and finite index image. This theorem will be restated later with a more quantitatively precise conclusion; see Theorem 9. Whyte is also able to apply his techniques to obtain a strong quasi-isometric rigidity theorem for the group Z ⋊GL(n,Z), which we will not state here. Our theorem about MCG(S1 g ), answers a special case of: Conjecture 2. If S is a nonexceptional surface of finite type then for any finitely generated group K quasi-isometric to MCG(S) there exists a homomorphism K → MCG(S) with finite kernel and finite index image.