Metric distortion in the geometric Schottky problem

The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus J(Mg) in the moduli space Ag of principally polarized abelian varieties as an algebraic subvariety. By viewing Ag as a noncompact metric space coming from its structure as a locally symmetric space and J(Mg) as a metric subspace, we compare the subspace metric d and the induced length metric ? on J(Mg). Consequently, we clarify the nature of the metric distortion of the subspace J(Mg) and hence settle a problem posed by Farb(2006) on the metric distortion of J(Mg) inside Ag in a certain sense(see Theorem 1.5 and Corollary 1.6).