We exhibit a group of type F whose second cohomology contains a weakly bounded, but not bounded, class. As an application, we disprove a long-standing conjecture of Gromov about bounded primitives of differential forms on universal covers of closed manifolds.
We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and—for almost all primes p—(mathbb{F} _{p})-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic 7-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.
We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows.
A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.
Avila’s Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency (SL(2,{mathbb{R}})) cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila’s, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case.
We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let (X,Y subset mathbb{R}^{2}) be non-empty Borel sets. If X is not contained in any line, we prove that
$$ sup _{x in X} dim _{mathrm {H}}pi _{x}(Y , setminus , {x}) geq min { dim _{mathrm {H}}X,dim _{mathrm {H}}Y,1}. $$If dimHY>1, we have the following improved lower bound:
$$ sup _{x in X} dim _{mathrm {H}}pi _{x}(Y , setminus , {x}) geq min { dim _{mathrm {H}}X + dim _{mathrm {H}}Y - 1,1}. $$Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck’s theorem in combinatorial geometry: if (X subset mathbb{R}^{2}) is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines (ell subset mathbb{R}^{2}), then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}.
While the results above concern (mathbb{R}^{2}), we also derive some counterparts in (mathbb{R}^{d}) by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren.
We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin. (3) The parallelizability of planar d-webs. (4) The Elekes-Rónyai theorem on expanding polynomials.
Given a Borel set A in the plane, we study the set of exceptional vantage points, for which the pinned distance Δp(A) has small dimension, that is, close to (dimA)/2. We show that if this set has positive dimension, then it must have very special structure. This result follows from a more general single-scale nonlinear projection theorem, which says that if ϕ1, ϕ2, ϕ3 are three smooth functions whose associated 3-web has non-vanishing Blaschke curvature, and if A is a (δ,α)2-set in the sense of Katz and Tao, then at least one of the images ϕi(A) must have measure much larger than |A|1/2, where |A| stands for the measure of A. We prove analogous results for d smooth functions ϕ1,…,ϕd, whose associated d-web is not parallelizable.
We use similar tools to characterize when bivariate real analytic functions are “dimension expanding” when applied to a Cartesian product: if P is a bivariate real analytic function, then P is either locally of the form h(a(x)+b(y)), or P(A,B) has dimension at least α+c whenever A and B are Borel sets with Hausdorff dimension α. Again, this follows from a single-scale estimate, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture.
A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called Morse flats. We show that the Tits boundary ∂TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂TF. More precisely, we show that the set of singular points in ∂TF can be covered by finitely many round spheres of positive codimension.
We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of (mathbb{R}^{n}) . We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.
We give an optimal bound on normal curvatures of immersed n-torus in a Euclidean ball of large dimension.
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