In this paper we start the classification of strongly bracket-generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.

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We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The limit for coordinate spheres of a boosted slice of the Schwarzchild solution is computed explicitly and shown to give the expected energymomentum four-vector

Let .. be the fundamental group of a compact riemannian manifold X of sectional curvature K ≤ .1 and dimension n ≥ 3. We suppose that .. = A .C B is the free product of its subgroups A and B amalgamated over the subgroup C. We prove that the critical exponent δ(C) of C satisfies δ(C) ≥ n.2. The equality occurs if and only if there exist an embedded compact hypersurface Y . X, totally geodesic, of constant sectional curvature .1, whose fundamental group is C and which separates X in two connected components whose fundamental groups are A and B respectively. Similar results hold if .. is an HNN extension, or more generally if .. acts on a simplicial tree without fixed point.

We state and prove a long-elusive relation between genus-one Gromov-Witten of a complete intersection and twisted Gromov Witten invariants of the ambient projective space. As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component M 0 1,k(Pn, d) of the moduli space of stable genus-one holomorphic maps into Pn have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, J-holomorphic maps. This extension is used to show that these cones are the correct genus-one analogues of the vector bundles relating genuszero Gromov-Witten invariants of a complete intersection to those of the ambient projective space. A relationship for higher-genus invariants is conjectured as well.