We consider the stable norm associated to a discrete, torsionless abelian group of isometries Γ ∼ = Z n of a geodesic space (X, d). We show that the difference between the stable norm st and the distance d is bounded by a constant only depending on the rank n and on upper bounds for the diameter ofX = Γ\X and the asymptotic volume ω(Γ, d). We also prove that the upper bound on the asymptotic volume is equivalent to a lower bound for the stable systole of the action of Γ on (X, d); for this, we establish a Lemmaà la Margulis for Z n -actions, which gives optimal estimates of ω(Γ, d) in terms of stsys(Γ, d), and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters n, diam(X) and ω(Γ, d) (or stsys (Γ, d)) are necessary to bound the difference d − st , by providing explicit counterexamples for each case. As an application in Riemannian geometry, we prove that the number of connected components of any optimal, integral 1-cycle in a closed Riemannian manifoldX either is bounded by an explicit function of the first Betti number, diam(X) and ω(H 1 (X, Z)), or is a sublinear function of the mass.
We study Riemannian metrics on compact, torsionless, non-geometric 3-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "à la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then deduce corresponding local topological rigidy results in the class M ∂ ngt (E, D) of compact non-geometric 3-manifolds with torsionless fundamental group (with possibly empty, non-spherical boundary) whose entropy and diameter are bounded respectively by E, D. For instance, this class locally contains only finitely many topological types; and closed, irreducible manifolds in this class which are close enough (with respect to E, D) are diffeomorphic. Several examples and counter-examples are produced to stress the differences with the geometric case.
We give a sharp comparison between the spectra of two Riemannian manifolds (Y, g) and (X, g0) under the following assumptions: (X, g0) has bounded geometry, (Y, g) admits a continuous Gromov-Hausdorff ε-approximation onto (X, g0) of non zero absolute degree, and the volume of (Y, g) is almost smaller than the volume of (X, g0). These assumptions imply no restrictions on the local topology or geometry of (Y, g) in particular no curvature assumption is supposed or infered.
We prove that there exists a positive, explicit function F (k, E) such that, for any group G admitting a k-acylindrical splitting and any generating set S of G with Ent(G, S) < E, we have |S| ≤ F (k, E). We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, D-quasiconvex k-malnormal amalgamated products acting on δ-hyperbolic spaces or on CAT (0)-spaces with entropy bounded by E. A number of finiteness results for interesting families of Riemannian or metric spaces with bounded entropy and diameter also follow: Riemannian 2-orbifolds, non-geometric 3-manifolds, higher dimensional graph manifolds and cusp-decomposable manifolds, ramified coverings and, more generally, CAT(0)-groups with negatively curved splittings. Contents 24 2.4. Non-geometric 3-manifolds 27 2.5. Ramified coverings 28 2.6. Higher dimensional graphs and cusp decomposable manifolds 29 Appendix A. Acylindrical splittings of hyperbolic 2-orbifolds 30 Appendix B. 3-manifolds with prescribed fundamental group 31 Appendix C. Malnormal subgroups of CAT(0)-groups 34 References 35
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