This thesis deals with global Riemannian geometry without curvature assumptions and its link to topology, we focus on the maximal volume of balls of fixed radius in the universal covers of graphs and surfaces. In the first part, we prove that if the area of a closed Riemannian surface M of genus at least two is sufficiently small with respect to its hyperbolic area, then for every radius R>0 the universal cover of M contains an R-ball with area at least the area of a cR-ball in the hyperbolic plane, where c0 the universal cover of Gamma contains an R-ball with length at least c times the length of an R-ball in the universal cover of Gamma_b, where c is in the interval (1/2,1) is a universal constant. In the second part, we generalize a theorem of M. Gromov concerning the maximal number of homotopically independentshort loops based at the same point . Specifically, we prove that on every closed Riemannian surface M of genus g and area normalized to g there exist at least log(2g) homotopically independent loops based at the same point of length at most C log(g), where C is some positive constant independent from the genus. As an immediate corollary of this theorem, we recapture the asymptotic systolic inequality on the separating systole. We also prove a similar theorem for metric graphs. Precisely, we prove that on every metric graph Gamma of first Betti number and length b, there exist at least log(b) homologically independent loops based at the same point of length at most 48 log(b). That extends Bollobàs-Szemerédi-Thomason's log(b) bound on the homological systole to at least log(b) homologically independent loops based at the same point. Moreover, we give examples of graphs where our result is optimal (up to a multiplicative constant).
