On the fundamental group of Riemannian manifolds with nonnegative Ricci curvature (Q1938043)

The Bonnet-Myers theorem implies that manifolds of strictly positive (bounded away from zero) Ricci curvature have finite fundamental group. This holds more generally for manifolds of nonnegative Ricci curvature under the assumption \(\limsup_{R\rightarrow\infty}\frac{\mathrm{vol}(B_p(R))}{R^n}>0\), as shown by \textit{P. Li} [Ann. Math. (2) 124, 1--21 (1986; Zbl 0613.58032)] and \textit{M. Anderson} [Topology 29, No. 1, 41--55 (1990; Zbl 0696.53027)]. A conjecture of Milnor asserts that manifolds of nonpositive Ricci curvature should at least have finitely generated fundamental group. This was proved by \textit{C. Sormani} [J. Differ. Geom. 54, No. 3, 547--559 (2000; Zbl 1035.53045)] in the case \(\limsup_{R\rightarrow\infty}\frac{\mathrm{diam}(\partial B_p(R))}{R}<4S_n\) for some explicit \(S_n\). The paper under review proves another special case of Milnors conjecture, in generalizing the result of Li and Anderson: for any \(n\in \mathbb N\), there exists \(\epsilon>0\) such that, if \(M\) is an \(n\)-dimensional manifold of nonpositive Ricci curvature and \(\lim_{R\rightarrow\infty}\frac{\mathrm{vol}(B_p(R))}{R^{n-\alpha}}>0\) for some \(\alpha\in[0,\epsilon)\), then \(\pi_1M\) is finitely generated. The proof uses the Bishop-Gromov comparison theorem and the generators of the fundamental group constructed by \textit{N. Yeganefar} [Differ. Geom. Appl. 25, No. 3, 251--257 (2007; Zbl 1141.53038)].