Systolic Geometry - Related Fields, Volume Entropy

Related Fields, Volume Entropy

Asymptotic phenomena for the systole of surfaces of large genus have been shown to be related to interesting ergodic phenomena, and to properties of congruence subgroups of arithmetic groups.

Gromov's 1983 inequality for the homotopy systole implies, in particular, a uniform lower bound for the area of an aspherical surface in terms of its systole. Such a bound generalizes the inequalities of Loewner and Pu, albeit in a non-optimal fashion.

Gromov's seminal 1983 paper also contains asymptotic bounds relating the systole and the area, which improve the uniform bound (valid in all dimensions).

It was discovered recently (see paper by Katz and Sabourau below) that the volume entropy h, together with A. Katok's optimal inequality for h, is the "right" intermediary in a transparent proof of M. Gromov's asymptotic bound for the systolic ratio of surfaces of large genus.

The classical result of A. Katok states that every metric on a closed surface M with negative Euler characteristic satisfies an optimal inequality relating the entropy and the area.

It turns out that the minimal entropy of a closed surface can be related to its optimal systolic ratio. Namely, there is an upper bound for the entropy of a systolically extremal surface, in terms of its systole. By combining this upper bound with Katok's optimal lower bound in terms of the volume, one obtains a simpler alternative proof of Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, such an approach yields an improved multiplicative constant in Gromov's theorem.

As an application, this method implies that every metric on a surface of genus at least 20 satisfies Loewner's torus inequality. This improves the best earlier estimate of 50 which followed from an estimate of Gromov's.