Given a convex representation ρ:Γ→PGL(d,R)\documentclass[12pt]{minimal}
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\begin{document}$$\rho :\Gamma \rightarrow {{\mathrm{PGL}}}(d,\mathbb {R})$$\end{document} of a convex cocompact group Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} of Isom+Hk,\documentclass[12pt]{minimal}
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\begin{document}$${{\mathrm{Isom}}}_+\mathbb {H}^k,$$\end{document} we find upper bounds for the quantity αhρ,\documentclass[12pt]{minimal}
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\begin{document}$$\alpha h_\rho ,$$\end{document} where hρ\documentclass[12pt]{minimal}
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\begin{document}$$h_\rho $$\end{document} is the entropy of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} and α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is the Hölder exponent of the equivariant map ∂∞Γ→P(Rd).\documentclass[12pt]{minimal}
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\begin{document}$${{\partial }_{\infty }}\Gamma \rightarrow \mathbb {P}(\mathbb {R}^d).$$\end{document} We also give rigidity statements when the upper bound is attained. This provides an analog of Thurston’s metric for convex cocompact groups of Isom+Hk.\documentclass[12pt]{minimal}
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\begin{document}$${{\mathrm{Isom}}}_+\mathbb {H}^k.$$\end{document} We then prove that if ρ:π1Σ→PSL(d,R)\documentclass[12pt]{minimal}
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\begin{document}$$\rho :\pi _1\Sigma \rightarrow {{\mathrm{PSL}}}(d,\mathbb {R})$$\end{document} is in the Hitchin component then αhρ≤2/(d-1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha h_\rho \le 2/(d-1)$$\end{document} (where α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is the Hölder exponent of Labourie’s equivariant flag curve) with equality if and only if ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} is Fuchsian.
