No abstract

We study the fundamental group of an open n-manifold M of nonnegative Ricci curvature with additional stability condition on M , the Riemannian universal cover of M . We prove that if any tangent cone of M at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then π 1 (M ) is finitely generated and contains a normal abelian subgroup of finite index; if in addition M has Euclidean volume growth of constant at least L, then we can bound the index of that abelian subgroup in terms of n and L. In particular, our result implies that if M has Euclidean volume growth of constant at least 1 − ǫ(n), then π 1 (M ) is finitely generated and C(n)-abelian.

Let X be a non-collapsing Ricci limit space and let x ∈ X. We show that for any ǫ > 0, there is r > 0 such that every loop in Bt(x) is contractible in B (1+ǫ)t (x), where t ∈ (0, r]. In particular, X is semi-locally simply connected.

Let M be an open n-manifold of nonnegative Ricci curvature and let p ∈ M {p\in M} . We show that if ( M , p ) {(M,p)} has escape rate less than some positive constant ϵ ⁢ ( n ) {\epsilon(n)} , that is, minimal representing geodesic loops of π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} escape from any bounded balls at a small linear rate with respect to their lengths, then π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} is virtually abelian. This generalizes the author’s previous work [J. Pan, On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature, Geom. Topol. 25 2021, 2, 1059–1085], where the zero escape rate is considered.

Let M be an open n-manifold with nonnegative Ricci curvature. We prove that if its escape rate is not 1/2 and its Riemannian universal cover is conic at infinity, that is, every asymptotic cone (Y, y) of the universal cover is a metric cone with vertex y, then π 1 (M ) contains an abelian subgroup of finite index. If in addition the universal cover has Euclidean volume growth of constant at least L, we can further bound the index by a constant C(n, L).We study the virtual abelianness/nilpotency of fundamental groups of open manifolds with Ric ≥ 0. According to the work of Kapovitch-Wilking [7], these fundamental groups always have nilpotent subgroups with index at most C(n) (also see [8,6]). In general, these fundamental groups may not contain any abelian subgroups with finite index, because Wei has constructed examples with torsion-free nilpotent fundamental groups [15]. This is different from manifolds with sec ≥ 0, whose fundamental groups are always virtually abelian [3].A question raised from here is, for an open manifold M with Ric ≥ 0, on what conditions is π 1 (M ) virtually abelian? To answer this question, one naturally looks for indications from the geometry of nonnegative sectional curvature. Ideally, if the manifold M fulfills some geometric conditions modeled on nonnegative sectional curvature, even in a much weaker form, then π 1 (M ) may turn out to be virtually abelian. In other words, when π 1 (M ) is not virtually abelian, some aspects of M should be drastically different from the geometry of nonnegative sectional curvature.We have explored this direction in [11,12], from the viewpoint of escape rate. Recall that each element γ in π 1 (M, p) can be represented by a geodesic loop at p, denoted by c γ , with the minimal length in its homotopy class. If M has sec ≥ 0, then all these representing loops must stay in a bounded ball; however, this property in general does not hold for nonnegative Ricci curvature. The escape rate measures how fast these loops escape from bounded balls: