Gromov's Theorem Research Articles

Ricci curvature of metric spaces

We define a notion of Ricci curvature in metric spaces equipped with a measure or a random walk. For this we use a local contraction coefficient of the random walk acting on the space of probability measures equipped with a transportation distance. This notions allows to generalize several classical theorems associated with positive Ricci curvature, such as a spectral gap bound (Lichnerowicz theorem), Gaussian concentration of measure (Lévy–Gromov theorem), logarithmic Sobolev inequalities (a result of Bakry–Émery theory) or the Bonnet–Myers theorem. The definition is compatible with Bakry–Émery theory, and is robust and very easy to implement in concrete examples such as graphs. To cite this article: Y. Ollivier, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

The main result of this paper is that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not Kahler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2.

An orbifold relative index theorem

In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.

SOME SMALL CANCELLATION PROPERTIES OF RANDOM GROUPS

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant 1-2d depending upon the density parameter d. This implies in particular a property generalizing the ordinary C′ small cancellation condition, which could be termed "macroscopic cancellation". This also sharpens the evaluation of the hyperbolicity constant δ. As a consequence we get that the standard presentation of a random group at density d < 1/5 satisfies the Dehn algorithm and Greendlinger's lemma, and that it does not for d > 1/5. For this we establish a version of the local-global principle for hyperbolic spaces (Cartan–Hadamard–Gromov theorem) involving arbitrarily small loss in the isoperimetric constant.

Adding high powered relations to large groups

It is a common theme in group theory that if a group G is quotiented by a sufficiently high power of an element g ∈ G, then properties of G are inherited by the quotient group. A major example of this phenomenon is the theorem of Gromov ([4], see also [2]) asserting that if G is torsion-free and word-hyperbolic, then G/〈〈g〉〉 is also word-hyperbolic for all sufficiently big integers n. In this paper, we will prove that this phenomenon occurs for the property of being large. Recall that G is large if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. Large groups have many interesting and useful properties, including super-exponential subgroup growth and infinite virtual first Betti number. They are also particularly important in low-dimensional topology. Our main theorem is the following.

On a small cancellation theorem of Gromov

We give a combinatorial proof of a theorem of Gromov, which extends the scope of small cancellation theory to group presentations arising from labelled graphs.

A curvature-dimension condition for metric measure spaces

We present a curvature-dimension condition CD ( K , N ) for metric measure spaces ( M , d , m ) . In some sense, it will be the geometric counterpart to the Bakry–Émery [D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités XIX, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. [1]] condition for Dirichlet forms. For Riemannian manifolds, it holds if and only if dim ( M ) ⩽ N and Ric M ( ξ , ξ ) ⩾ K ⋅ | ξ | 2 for all ξ ∈ TM . The curvature bound introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Math., in press. [4]; K.T. Sturm, Generalized Ricci bounds and convergence of metric measure spaces, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 235–238. [6]; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., in press. [7]] is the limit case CD ( K , ∞ ) . Our curvature-dimension condition is stable under convergence. Furthermore, it entails various geometric consequences e.g. the Bishop–Gromov theorem and the Bonnet–Myers theorem. In both cases, we obtain the sharp estimates known from the Riemannian case. To cite this article: K.-T. Sturm, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

DIRECTED EMBEDDINGS OF CLOSED MANIFOLDS

Gromov's theorem on directed embeddings of open manifolds is generalized to the case of closed manifolds, where the embeddings are directed with respect to a given codimension ≥ 1 foliation on the manifold. Applications include directed embeddings with prescribed codimension 1 cusp singularities.

Notes on symplectic non-squeezing of the KdV flow

We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle 𝕋. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.

The compression theorem III: applications

This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts (math.GT/9712235 and math.GT/0003026) gave proofs. Here we are concerned with applications. We give short new (and constructive) proofs for immersion theory and for the loops--suspension theorem of James et al and a new approach to classifying embeddings of manifolds in codimension one or more, which leads to theoretical solutions. We also consider the general problem of controlling the singularities of a smooth projection up to C^0--small isotopy and give a theoretical solution in the codimension > 0 case.

Cellular automata and strongly irreducible shifts of finite type

If A is a finite alphabet and Γ is a finitely generated amenable group, Ceccherini-Silberstein, Machı̀ and Scarabotti have proved that a local transition function defined on the full shift AΓ is surjective if and only if it is pre-injective; this equivalence is the so-called Garden of Eden theorem. On the other hand, when Γ is the group of the integers, the theorem holds in the case of irreducible shifts of finite type as a consequence of a theorem of Lind and Marcus but it no longer holds in the two-dimensional case.Recently, Gromov has proved a GOE-like theorem in the much more general framework of the spaces of bounded propagation. In this paper we apply Gromov's theorem to our class of spaces proving that all the properties required in the hypotheses of this theorem are satisfied.We give a definition of strong irreducibility that, together with the finite-type condition, it allows us to prove the GOE theorem for the strongly irreducible shifts of finite type in AΓ (provided that Γ is amenable). Finally, we prove that the bounded propagation property for a shift is strictly stronger than the union of strong irreducibility and finite-type condition.

The compression theorem I

This the first of a set of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer-Verlag (1986); 2.4.5 C'] and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding. In the second paper in the series we give a proof in the spirit of Gromov's proof and in the third part we give applications.

The compression theorem II: directed embeddings

This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.

Spherical space forms and Dehn filling

This paper concerns those Dehn fillings on a torally bounded 3-manifold which yield manifolds with a finite fundamental group. The focus will be on those torally bounded 3-manifolds which either contain an essential torus, or whose interior admits a complete hyperbolic structure. While we give several general results, our sharpest theorems concern Dehn fillings on manifolds which contain an essential torus. One of these results is a sharp “finite surgery theorem.” The proof incl udes a characterization of the finite fillings on “generalized” iterated torus knots with a complete classification for the iterated torus knots in the 3-sphere. We also give a proof of the so-called “2π” theorem of Gromov and Thurston, and obtain an improvement (by a factor of two) in the original estimates of Thurston on the number of non-negatively-curved Dehn fillings on a torally bounded 3-manifold whose interior admits a complete hyperbolic structure.

Quasiconvex subgroups of negatively curved groups

If H is a finitely generated group, then Γ〈 h 1,…, h n 〉, the Cayley graph of H with respect to a finite generating set { h 1,…, h n }, has as vertices the elements of H. There is an edge between the vertices v and w of Γ if vh i = w for some i ∈ {1,…, n}. Theorem. Let H be a negatively curved group. If A is an infinite quasiconvex subgroup of H (i.e. there is a real number ε such that every geodesic in Γ between points of A is within ε of A) then: (1) A has finite index in the normalizer of A in H. (2) If h ε H and hAh -1 is a subset of A then hAh -1 = A. (3) If N is an infinite normal subgroup of H and N ⊂ A, then A has finite index in H. In a negatively curved group, infinite cyclic subgroups are quasiconvex. Hence (1) generalizes a theorem of Gromov's that centralizers of elements of infinite order in a negatively curved group are virtually cyclic. Finitely generated free groups are negatively curved and any finitely generated subgroup of a finitely generated free group is quasiconvex. The special case of (3) where H is free and A is finitely generated is well known.

Moderate growth and random walk on finite groups

We study the rate of convergence of symmetric random walks on finite groups to the uniform distribution. A notion of moderate growth is introduced that combines with eigenvalue techniques to give sharp results. Roughly, for finite groups of moderate growth, a random walk supported on a set of generators such that the diameter of the group is γ requires order γ2 steps to get close to the uniform distribution. This result holds for nilpotent groups with constants depending only on the number of generators and the class. Using Gromov's theorem we show that groups with polynomial growth have moderate growth.

Relative index theory

We develop the framework for the heat equation method in the relative index theory. We study pairs of Dirac operators on complete manifolds and give an analytic interpretation of the difference of the integrals over their local index densities. This yields a generalization of the relative index theorem of Gromov and Lawson. Under some assumptions on the curvature we show, that supersymmetric scattering theories of Borisov, Müller, and Schrader arise naturally. The scattering index is related to the relative topological index.

Local Rigidity of Symmetric Spaces

We show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds, then the new metric is isometric to the old one. We also prove an analytic continuation property for symmetric spaces of rank > 3 . The aim of this paper is to show that a symmetric space is rigid under a local modification of the metric which preserves suitable curvature bounds. Our results were motivated by the following theorem of Gromov [BGS, ?5]: Theorem 1 (Gromov). Let X be a simply connected symmetric space of rank > 2 and with nonpositive sectional curvature and Q the complement of a compact set in X. If f: Q -Y is a totally geodesic isometric embedding, where Y is a complete simply connected Riemannian manifold with nonpositive sectional curvature, then there exists a unique extension of f to a totally geodesic isometric embedding Tf: X-Y. We will give a slightly different proof of this theorem, which will then enable us to also prove the following dual result: Theorem 2. Let X be a simply connected symmetric space of rank > 2 and with sectional curvature 0 0, and dim Y = dim X, and if Q is the complement of a ball of radius 7r/30 in X, then any isometric embedding f: Q -Y extends uniquely to an isometry 7: X -Y. If in the above two theorems, we choose Y = X and f = id, we obtain the desired local rigidity of symmetric spaces. Theorem 2 applies in particular to S2 x S and shows that the Hopf conjecture is at least locally correct. The optimal radius for the ball in Theorem 2 would be the radius of a convex ball (> 7r/2 under those assumptions). Indeed, in the case of nonpositive curvature, Gromov [BGS, ?5] proved the following local version of Theorem 1: Received by the editors November 4, 1987 and, in revised form, September 12, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C35; Secondary 53C20. The first author is supported by the Schweizerischer Nationalfond. The second author is partially supported by a grant from the National Science Foundation and would like to thank the IHES and the Max Planck Institut for its hospitality. (?) 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page

The C1,1 conclusions in Gromov's theory

AbstractAccording to M. Gromov, any sequence of Riemann manifolds with uniformly bounded geometry has a subsequence that converges to a limit. It is shown here that this limit Riemann structure is Lipschitz, generates a Lipschitz geodesic flow, and consequently, as Gromov asserted, the limit distance function is of class C1,1. Sharpness of the results is discussed. A simple, extrinsic proof of Gromov's Theorem is included.

Higher-order nondegenerate immersions of manifolds

In this paper we show how the Smale-Hirsch theory of immersions can be adapted to get a simple proof of an integrability theorem of Gromov and Eliashberg concerning higher-order nondegenerate immersions.