A geometric criterion on embedded disks along Teichmuller geodesics implies that for uniquely ergodic translation surfaces, almost every branched N-cover via a slit is uniquely ergodic.
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Separated families of Anosov representations have critical exponents asymptotic to a combinatorial invariant computable from finite graph spectral data, yielding bounds on the Thurston asymmetric metric and analysis of convex projective degenerations on a pair of pants.
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Unique ergodicity of branched covers of translation surfaces
A geometric criterion on embedded disks along Teichmuller geodesics implies that for uniquely ergodic translation surfaces, almost every branched N-cover via a slit is uniquely ergodic.
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On separated families of Anosov representations
Separated families of Anosov representations have critical exponents asymptotic to a combinatorial invariant computable from finite graph spectral data, yielding bounds on the Thurston asymmetric metric and analysis of convex projective degenerations on a pair of pants.