The π1-extension of a finitely generated Veech group admits a cocompact isometric action on a hyperbolic space constructed by collapsing regions over the convex hull of the limit set, implying the extension is hierarchically hyperbolic.
Word hyperbolic extensions of surface groups
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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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Extensions of finitely generated Veech groups
The π1-extension of a finitely generated Veech group admits a cocompact isometric action on a hyperbolic space constructed by collapsing regions over the convex hull of the limit set, implying the extension is hierarchically hyperbolic.