Limits of almost homogeneous spaces and their fundamental groups

Abstract: We say that a sequence of proper geodesic spaces $X_n$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_n \leq \text{Iso}(X_n)$ with $\text{diam} (X_n/G_n)\to 0$ as $n \to \infty$. We show that if a sequence $(X_n,p_n)$ of pointed almost homogeneous spaces converges in the pointed Gromov--Hausdorff sense to a space $(X,p)$, then $X$ is a nilpotent locally compact group equipped with an invariant geodesic metric. Under the above hypotheses, we show that if $X$ is semi-locally-simply-connected, then it is a nilpotent Lie group equipped with an invariant sub-Finsler metric, and for $n$ large enough, $\pi_1(X) $ is a subgroup of a quotient of $ \pi_1(X_n) $.