Limits of almost homogeneous spaces and their fundamental groups

We say that a sequence of proper geodesic spaces X₍ consists of almost homogeneous spaces if there is a sequence of discrete groups of isometries G₍ Iso (X₍) with diam (X₍/G₍) 0 as n. We show that if a sequence (X₍, p₍) 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, ₁ (X) is a subgroup of a quotient of ₁ (X₍).