Weil–Petersson full-measure conjecture for large trace-growth surfaces

Establish, with the notation of Theorem D, the existence of ǫ>0 such that for sufficiently large genus g the Weil–Petersson volume proportion of the set V_g^{ǫ}—consisting of points in the moduli space M_g whose trace set has growth exceeding n^{2−ǫ}—is equal to 1, i.e., Vol_WP(V_g^{ǫ})/Vol_WP(M_g)=1 for all sufficiently large g.

Background

Theorem D shows that for any fixed ǫ>0, the Weil–Petersson measure of surfaces with trace growth exceeding n^{2−ǫ} tends to full measure as genus grows (the ratio approaches 1 as g→∞). The authors conjecture a stronger statement asserting eventual full measure—equal to 1—for sufficiently large genus.

This strengthens the asymptotic result by eliminating the limit and asserting stabilization of the full-measure property in the high-genus regime.