If S is a hyperbolic surface and S̊. The surface obtained from S by removing a point. The mapping class groups Mod(S) and Mod(S̊) fit into a short exact sequence 1→π1(S)→Mod(S̊)→Mod(S) →1 We give a new criterion formapping classes i. The kernel to be pseudo- Anosov usin. The geometry of hyperbolic 3-manifolds. Namely, we show that if M is an ε-thick hyperbolic manifold homeomorphic to S × R, then an element of π1(M) ≅ π1(S) represents a pseudo- Anosov element of Mod( . S) if its geodesic representative is "wide". We establish similar criteria where M is replaced with a coarsely hyperbolic surface bundle coming from a δ-hyperbolic surface-group extension.
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