Limits of almost homogeneous spaces and their fundamental groups

TL;DR

This paper investigates the limits of almost homogeneous spaces under Gromov--Hausdorff convergence, revealing they are nilpotent groups with specific geometric structures, and explores implications for their fundamental groups.

Contribution

It characterizes the Gromov--Hausdorff limits of almost homogeneous spaces as nilpotent groups with invariant metrics, extending understanding of their geometric and topological properties.

Findings

Limits are nilpotent locally compact groups with invariant geodesic metrics.

If the limit space is semi-locally simply connected, it is a nilpotent Lie group with a sub-Finsler metric.

Fundamental groups of the spaces relate via subgroups and quotients for large n.

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) $.