Gromov's Theorem Research Articles

Three-Dimensional Graph Products with Unbounded Stack-Number

We prove that the stack-number of the strong product of three n-vertex paths is $$\Theta (n^{1/3})$$ . The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest $$\Delta _0$$ such that there exists a graph family with unbounded stack-number, bounded queue-number and maximum degree $$\Delta _0$$ . We show that $$\Delta _0\in \{6,7\}$$ .

Homology bounds for hyperbolic orbifolds

We will provide bounds on both the Betti numbers and the torsion part of the homology of hyperbolic orbifolds. These bounds are linear in the volume and are a direct consequence of an efficient simplicial model of the thick part, which we will construct as well. The homology statements complement previous work of Bader, Gelander and Sauer (torsion homology of manifolds), Samet (Betti numbers of orbifolds) and a classical theorem of Gromov (Betti numbers of manifolds). For arithmetic, non-compact hyperbolic orbifolds—i.e. in the case of arithmetic, non-uniform lattices in {text {Isom}}(mathbb {H}^n)—the strongest results will be obtained.

On the Pila–Wilkie theorem

This expository paper gives an account of the Pila–Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin–Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov.

A virtually 2-step nilpotent group with polynomial geodesic growth

A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.

Quantitative aspects of acyclicity

The Cheeger constant is a measure of the edge expansion of a graph and as such plays a key role in combinatorics and theoretical computer science. In recent years, there is an interest in k-dimensional versions of the Cheeger constant that likewise provide quantitative measure of cohomological acyclicity of a complex in dimension k. In this paper, we study several aspects of the higher Cheeger constants. Our results include methods for bounding the cosystolic norm of k-cochains and the k-th Cheeger constants, with applications to the expansion of pseudomanifolds, Coxeter complexes and homogenous geometric lattices. We revisit a theorem of Gromov on the expansion of a product of a complex with a simplex and provide an elementary derivation of the expansion in a hypercube. We prove a lower bound on the maximal cosystole in a complex and an upper bound on the expansion of bounded degree complexes and give an essentially sharp estimate for the cosystolic norm of the Paley cochains. Finally, we discuss a non-abelian version of the one-dimensional expansion of a simplex, with an application to a question of Babson on bounded quotients of the fundamental group of a random 2-complex.

A functional analysis proof of Gromov's polynomial growth theorem

The celebrated theorem of Gromov asserts that any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner and others. In this note, we give yet another proof of Gromov's theorem, along the lines of Shalom and Chifan--Sinclair, which is based on the analysis of reduced cohomology and Shalom's property H_FD.

On the scaling limit of finite vertex transitive graphs with large diameter

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.

A Gromov–Winkelmann type theorem for flexible varieties

An affine variety X of dimension ≥ 2 is called flexible if its special automorphism group SAut( X ) acts transitively on the smooth locus X_{\mathrm {reg}} [1]. Recall that SAut( X ) is the subgroup of the automorphism group Aut( X ) generated by all one-parameter unipotent subgroups [1]. Given a normal, flexible, affine variety X and a closed subvariety Y in X of codimension at least 2, we show that the pointwise stabilizer subgroup of Y in the group SAut( X ) acts infinitely transitively on the complement X \setminus Y , that is, m -transitively for any m ≥ 1 . More generally we show such a result for any quasi-affine variety X and codimension ≥ 2 subset Y of X . In the particular case of X = \mathbb A^n , n ≥2 , this yields a Theorem of Gromov and Winkelmann [8], [18].

Nilprogressions and groups with moderate growth

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter. We also give bounds on the Cheeger constant, first eigenvalue of the Laplacian, and mixing time. This can be seen as a finite-group version of Gromov's theorem on groups with polynomial growth. It also improves on a result of Lackenby regarding property (τ) in towers of coverings. Another consequence is a universal upper bound on the diameter of all finite simple groups, independent of the CFSG.

Positive Curvature, Partial Vanishing Theorems and Coarse Indices

AbstractWe generalize the relative index theorem of Gromov and Lawson, using coarse geometry.

Yang–Mills theory and jumping curves

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss the role of infinite-dimensional non-squeezing results in the study of Hamiltonian PDEs and show some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

The projective plane, J-holomorphic curves and Desarguesʼ theorem

By a theorem of Gromov, for an almost complex structure J on CP2 tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys.

Proof of Gromov’s theorem on homogeneous nilpotent approximation for vector fields of class C 1

The article is devoted to the asymptotic properties of the vector fields \(\tilde X_i^g \), i = 1, …, N, θg-connected with C1-smooth basis vector fields {Xi}i=1,…,N satisfying condition (+ deg). We prove a theorem of Gromov on the homogeneous nilpotent approximation for vector fields of classC1. Nontrivial examples are constructed of quasimetrics induced by vector fields {Xi}i=1, …, N.

Waist of the sphere for maps to manifolds

We generalize the sphere waist theorem of Gromov and the Borsuk–Ulam type measure partition lemma of Gromov–Memarian for maps to manifolds.

Discrete homotopies and the fundamental group

We generalize and strengthen the theorem of Gromov that the fundamental group of any compact Riemannian manifold of diameter at most D has a set of generators g1,…,gk of length at most 2D and relators of the form gigm=gj. In particular, we obtain an explicit bound for the number k of generators in terms of the number of “short loops” at every point and the number of balls required to cover a given semi-locally simply connected geodesic space. As a corollary we obtain a fundamental group finiteness theorem (new even for Riemannian manifolds) that replaces the curvature and volume conditions of Anderson and the 1-systole bound of Shen–Wei, by more general geometric hypothesis implied by these conditions. This theorem, in turn, is a special case of a theorem for arbitrary compact geodesic spaces, proved using the method of discrete homotopies introduced by the first author and V. N. Berestovskii. Central to the proof is the notion of “homotopy critical spectrum”, introduced in this paper as a natural consequence of discrete homotopy methods. This spectrum is closely related to the Sormani–Wei covering spectrum which is a subset of the classical length spectrum studied by de Verdiere and Duistermaat–Guillemin. It is completely determined (including multiplicity) by special closed geodesics called “essential circles”.

Approximate groups and doubling metrics

AbstractWe develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates.Our work complements three other recent approaches to developing non-abelian versions of Freĭman's theorem by Breuillard and Green, Fisher, Katz and Peng, and Tao.

A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem

A short and almost elementary proof of the Boros-F\"uredi-B\'ar\'any-Pach-Gromov theorem on the multiplicity of covering by simplices in $\mathbb R^d$ is given.

Fold-forms for four-folds

This paper explains an application of Gromov’s h-principle to prove the existence, on any orientable four-manifold, of a folded symplectic form. That is a closed two-form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic form by a folding map. We use the h-principle for folding maps (a theorem of Eliashberg) and the h-principle for symplectic forms on open manifolds (a theorem of Gromov) to show that, for orientable even-dimensional manifolds, the existence of a stable almost complex structure is necessary and sufficient to warrant the existence of a folded symplectic form.

Stable systolic category of manifolds and the cup-length

It follows from a theorem of Gromov that the stable systolic category \({\rm cat}_{\rm stsys} M\) of a closed manifold M is bounded from below by \({\rm cl}_{\mathbb{Q}} M\), the rational cup-length of M [Ka07]. We study the inequality in the opposite direction. In particular, combining our results with Gromov’s theorem, we prove the equality \({\rm cat}_{\rm stsys} M = {\rm cl}_{\mathbb{Q}} M\) for simply connected manifolds of dimension ≤ 7.