The lower semi-continuity of a fundamental group and
nilpotent structures in persistence.
Abstract: When a sequence of compact geodesic spaces Xi converges to a compact geodesic space X, under minimal assumptions there are surjective morphisms π1(Xi) → π1(X) for i large enough. In particular, a limit of simply connected spaces is simply connected. This is clearly not true for non-compact limits as one can see from a sequence of ellipsoids converging to a cylinder. We study how symmetries can allow one to study this lower semi-continuity of π1 in the non-compact case, and how nilpotent structures naturally arise in this context.