Asymptotic geometry of negatively curved manifolds of finite volume

We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $\Gamma$. If the quotient manifold $\bar X= \Gamma \backslash X$ is asymptotically $1/4$-pinched, we prove that $\Gamma$ is divergent and $U\bar X$ has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls $B(x,R)$ in $X$ is asymptotically equivalent to a purely exponential function $c(x)e^{\delta R}$, where $\delta$ is the topological entropy of the geodesic flow of $\bar X$. \linebreak This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices $\Gamma$ in negatively curved spaces $X$ (not asymptotically $1/4$-pinched) where, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen-Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential.Comment: 25 p. This paper replaces arXiv:1503.03971, withdrawn by the authors due to the Theorem 1.1 whose statement is far from the main subject of the paper; for the sake of clearness, this new version concentrates only on the question of volume growth (theorems 1.2, 1.3 and 1.4). Theorem 1.1 of arXiv:1503.03971 is now the subject of another paper (Signed only by 2 authors Sambusetti and Peign\'e) focused on this rigidity problem with a much better presentation of the context and another rigidity resul

Asymptotic geometry of negatively curved manifolds of finite volume

We study the asymptotic behavior of simply connected Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice Γ. If the quotient manifold X = Γ\X is asymptotically 1=4-pinched, we prove that Γ is divergent and U X has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls B(x,R) in X is asymptotically equivalent to a purely exponential function c.x/eδR, where δ is the topological entropy of the geodesic flow of X . This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices Γ in negatively curved spaces X (not asymptotically 1/4-pinched) where, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen- Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential

Entropies, volumes, and Einstein metrics

We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstruction to the existence of Einstein metrics on four-manifolds involving the volume entropy. This generalizes both the Gromov--Hitchin--Thorpe inequality and Sambusetti's obstruction.Comment: This is a substantial revision and expansion of the 2004 preprint, which I prepared in spring of 2010 and which has since been published. The version here is essentially the published one, minus the problems introduced by Springer productio

On the growth of nonuniform lattices in pinched negatively curved manifolds

We study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold

Dominant Topologies in Euclidean Quantum Gravity

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according on the sign of the cosmological constant. For $\Lambda>0$, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For $\Lambda<0$, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the ``density of topologies'' grows fast enough to overwhelm this suppression. The value $\Lambda=0$ is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wave function.Comment: 14 pages, LaTeX. Minor additions (computability, relation to ``minimal volume'' in topology); error in eqn (3.5) corrected; references added. To appear in Class. Quant. Gra

Effective-Range Expansion of the Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential

The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.Comment: 21 pages, 6 figures and 6 tables, submitted to Prog. Theor. Phy

Growth tightness in group theory and Riemannian geometry

This paper is a survey on the "growth tightness" results for discrete groups and fundamental groups of negatively curved manifolds, based on original work of the author. We are interested in the exponential growth rate (also known as the entropy Ent(G)) of discrete group G, endowed with a left invariant metric; the metric may as well be a word metric or, when G acts freely by isometries on a Riemannian manifold X, the Riemannian distance between points of an orbit Gx (i.e. the Riemannian lenght of geodesic loops at x). We begin giving an elementary proof of growth tightness of free groups G, implying a well-known asymptotic characterization of free groups: for any group G on set A of k generators, we have Ent(G) < log(2k-1), with equality if and only if G is free on A. We also give in this case a precise estimate of the entropy gap Ent(G)-Ent(G/N) between the entropy of G and the entropy of any quotient G/N, in terms of the length of the relations. In the second part, we give similar statements for free and amalgamated products, introducing and estimating the entropy of the space Lc(G) of "weighted words" on G, to circumvent the difficulty of working directly with minimal reduced words in the general case. Section 3 contains the analogue of these results for fundamental groups G of a compact closed hyperbolic surface S, translating the growth tightness condition in terms of Galois coverings S' of the surface S; we also give estimates for the entropy gap Ent(H^2)-Ent(S') in terms of the systole of the covering S'. In section 4, we generalize these results to general, closed, negatively curved Riemannian manifolds

Packing and doubling in metric spaces with curvature bounded above

We study locally compact, locally geodesically complete, locally CAT(κ) spaces (GCBA κ-spaces). We prove a Croke-type local volume estimate only depending on the dimension of these spaces. We show that a local doubling condition, with respect to the natural measure, implies pure-dimensionality. Then we consider GCBA κ-spaces satisfying a uniform packing condition at some fixed scale r or a doubling condition at arbitrarily small scale, and prove several compactness results with respect to pointed Gromov–Hausdorff convergence. Finally, as a particular case, we study convergence and stability of Mκ-complexes with bounded geometry

Entropy Rigidity of negatively curved manifolds of finite volume

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2leq K_X leq -1$, then $Ent_top(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds