Volume growth and the topology of Gromov-Hausdorff limits

Abstract

We examine topological properties of pointed metric measure spaces $(Y, p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds $\{(M^n_i, p_i)\}_{i=1}^{\infty}$ with nonnegative Ricci curvature. Cheeger and Colding \cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $ν$ on the limit space $(Y, p)$. In the current work, we generalize previous results of the author to examine the relationship between the topology of $(Y, p)$ and the volume growth of $ν$. In particular, we prove a Abresch-Gromoll type excess estimate for triangles formed by limiting geodesics in the limit space. Assuming explicit volume growth lower bounds in the limit, we show that if $\lim_{r \to \infty} \frac{ν(B_p(r))}{ω_n r^n} > α(k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $α(k,n)$ are explicit and depend only on $n$, the dimension of the manifolds $\{(M^n_i, p_i)\}$, and $k$, the dimension of the homotopy in $(Y,p)$.