Determining surfaces by short curves and applications

Abstract The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $$g$$ g of a fixed arithmetic surface $$S$$ S are $$P(\frac{1}{g})$$ P ( 1 g ) apart from each other with respect to Teichmüller metric, where $$P$$ P is a polynomial depending only on $$S$$ S whose degree is universal. We also give a super-exponential upper bound for the number of semi-arithmetic hyperbolic surfaces with bounded genus, stretch and degree of the invariant trace field, generalizing to this class similar well-known bounds for arithmetic hyperbolic surfaces. In order to get these results we establish, for any closed hyperbolic surface $$S$$ S with injectivity radius at least $$s$$ s , a parametrization of the Teichmüller space by length functions whose values on $$S$$ S are bounded by a linear function (with constants depending only on $$s$$ s ) on the logarithm of the genus of $$S.$$ S .