Orbifolds with lower Ricci curvature bounds

We show that the first betti number $b_1(O)=\dim \,H_1(O,{\mathbb R})$ of a compact Riemannian orbifold $O$ with Ricci curvature $\operatorname {Ric}(O)\ge -(n-1)k$ and diameter $\operatorname {diam}(O)\le D$ is bounded above by a constant $c(n,kD^2)\ge 0$, depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the $b_1(O)$ is bounded above by the dimension $\dim \,O$, and that if, in addition, $b_1(O)=\dim \,O$, then $O$ is a flat torus $T^n$.