k-irreducible triangulations of orientable genus-g surfaces have O(k²g) triangles.
The geometry of Teichm¨ uller space via geodesic currents.Invent
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Conjecture that simple geodesics on finite modular covers have transcendental or algebraic endpoints, proved for minimal laminations.
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On the size of k-irreducible triangulations
k-irreducible triangulations of orientable genus-g surfaces have O(k²g) triangles.