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Limits of almost homogeneous spaces and their fundamental groups

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Zamora,  Sergio
Max Planck Institute for Mathematics, Max Planck Society;

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Zamora, S. (2024). Limits of almost homogeneous spaces and their fundamental groups. Groups, Geometry, and Dynamics, 18(3), 761-798. doi:10.4171/ggd/792.


Cite as: https://hdl.handle.net/21.11116/0000-000F-656E-5
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​).