Abstract
A Teichmüller lattice is the orbit of a point in Teichmüller space under the action of the mapping class group. We show that the proportion of lattice points in a ball of radius r which are not pseudo-Anosov tends to zero as r tends to infinity. In fact, we show that if R is a subset of the mapping class group, whose elements have an upper bound on their translation length on the complex of curves, then the proportion of lattice points in the ball of radius r which lie in R tends to zero as r tends to infinity.
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Maher, J. Asymptotics for Pseudo-Anosov Elements in Teichmüller Lattices. Geom. Funct. Anal. 20, 527–544 (2010). https://doi.org/10.1007/s00039-010-0064-9
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DOI: https://doi.org/10.1007/s00039-010-0064-9