Asymptotic bounds for separating systoles on surfaces

Abstract

The separating systole on a closed Riemannian surface M, denoted by sys0(M), is defined as the length of the shortest noncontractible loops which are homologically trivial. We answer positively a question of M. Gromov [Gr96, 2.C.2.(d)] about the asymptotic estimate on the separating systole. Specifically, we show that the separating systole of a closed Riemannian surface M of genus and area g satisfies an upper bound similar to M. Gromov’s asymptotic estimate on the (homotopy) systole. That is, sys0(M) is less than or equivalent to log g.