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Mathematics, Metric Geometry, Group Theory
Abstract:
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).
