Limits of almost homogeneous spaces and their fundamental groups
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) .
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