ASYMPTOTIC BOUNDS FOR SEPARATING SYSTOLES ON SURFACES

Abstract. The separating systole on a closed Riemannian sur-face M, denoted by sys 0 (M), is defined as the length of the short-est noncontractible loops which are homologically trivial. We an-swer 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. Gro-mov’s asymptotic estimate on the (homotopy) systole. That is, sys 0 (M). log g