Limits of almost homogeneous spaces and their fundamental groups

We say that a sequence of proper geodesic spaces Xₙ consists of almost homogeneous spaces if there is a sequence of discrete groups of isometries Gₙ ≤ Iso(Xₙ) with diam (Xₙ/Gₙ)→ 0 as n → ∞ . We show that if a sequence (Xₙ,pₙ) 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, π₁(X) is a subgroup of a quotient of π₁(Xₙ) .