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 $$g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> of a fixed arithmetic surface $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> are $$P(\frac{1}{g})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>g</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> apart from each other with respect to Teichmüller metric, where $$P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> is a polynomial depending only on $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> 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 $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> with injectivity radius at least $$s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> </mml:math> , a parametrization of the Teichmüller space by length functions whose values on $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> are bounded by a linear function (with constants depending only on $$s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> </mml:math> ) on the logarithm of the genus of $$S.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>