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 \(g\) of a fixed arithmetic surface \(S\) are \(P(\frac{1}{g})\) apart from each other with respect to Teichmüller metric, where \(P\) is a polynomial depending only on \(S\) 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 \(S\) with injectivity radius at least \(s\) , a parametrization of the Teichmüller space by length functions whose values on \(S\) are bounded by a linear function (with constants depending only on \(s\) ) on the logarithm of the genus of \(S.\)
