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Word hyperbolic extensions of surface groups

Ursula Hamenstaedt
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Abstract

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex cocompact in the sense of Farb and Mosher.

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Ursula Hamenstaedt. 2005-05-20. Word hyperbolic extensions of surface groups. https://arxiv.org/abs/math/0505244

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