Gromov-Hausdorff distances from simply connected geodesic spaces to the circle

Abstract

We prove that the Gromov-Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than $\frac{\pi }{4}$. We also prove that this bound is tight through the construction of a simply connected geodesic space $\mathrm{E}$ which attains the lower bound $\frac{\pi }{4}$. We deduce the first statement from a general result that we also establish which gives conditions on how small the Gromov-Hausdorff distance between two geodesic metric spaces $(X,d_X)$ and $(Y,d_Y)$ has to be in order for ${\pi }_1(X)$ and ${\pi }_1(Y)$ to be isomorphic.

15 pages, 6 figures