Fiber respecting quasi-isometries of surface group extensions
نویسنده
چکیده
Let S be a closed, oriented surface of genus g ≥ 2, and consider the extension 1 → π1S → MCG(S, p) → MCG(S) → 1, where MCG(S) is the mapping class group of S, and MCG(S, p) is the mapping class group of S punctured at p. We prove that any quasi-isometry of MCG(S, p) which coarsely respects the cosets of the normal subgroup π1S is a bounded distance from the left action of some element of MCG(S, p). Combined with recent work of Kevin Whyte this implies that ifK is a finitely generated group quasiisometric to MCG(S, p) then there is a homomorphism K → MCG(S, p) with finite kernel and finite index image. Our work applies as well to extensions of the form 1 → π1S → ΓH → H → 1, where H is an irreducible subgroup of MCG(S)—we give an algebraic characterization of quasi-isometries of ΓH that coarsely respect cosets of π1S.
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