Inventiones mathematicae最新文献

In this paper we confirm a folklore conjecture which suggests that for a complete noncompact manifold (M) of finite volume with sectional curvature (-1 leq K leq 0), if the universal cover of (M) is a visibility manifold, then the fundamental group of each end of (M) is almost nilpotent.

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor (beta >0) per edge. It arises as the (qto 0) limit of the (q)-state random cluster model with (p=beta q). We prove that in dimensions (dgeqslant 3) the arboreal gas undergoes a percolation phase transition. This contrasts with the case of (d=2) where no percolation transition occurs.

The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane (mathbb{H}^{0|2}). This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the (mathbb{H}^{0|2}) model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

Voiculescu discovered asymptotic freeness of independent Haar-distributed unitary matrices. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a nontrivial signature (rho ), i.e. two finite sequences of non-increasing natural numbers, and for (n) large enough, consider the irreducible representation (V_{n,rho }) of (mathbb{U}_{n}) associated with the signature (rho ). We consider the quotient (mathbb{U}_{n,rho }) of (mathbb{U}_{n}) viewed as a matrix subgroup of (mathbb{U}(V_{n,rho })), and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. To achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator-valued non-backtracking theory in a general (C^{*})-algebra, and finally, (iv) combinatorics of tensor moment matrices.

We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual (operatorname {CAT}(0)) cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber.

For the incompressible Navier-Stokes equations in (mathbb{R}^{3}) with low viscosity (nu >0), we consider the Cauchy problem with initial vorticity (omega _{0}) that represents an infinitely thin vortex filament of arbitrary given strength (Gamma ) supported on a circle. The vorticity field (omega (x,t)) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness (sqrt{nu t}) that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that (omega (x,t)) is well-approximated on a large time interval by (omega _{mathrm {lin}}(x-a(t),t)), where (omega _{mathrm {lin}}(cdot ,t)=exp (nu tDelta )omega _{0}) is the solution of the heat equation with initial data (omega _{0}), and (dot{a}(t)) is the instantaneous velocity given by Kelvin’s formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold’s geometric stability methods developed in the inviscid case (nu =0) to the slightly viscous situation. It turns out that although the geometric structures behind Arnold’s approach are no longer preserved by the equation for (nu > 0), the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold’s theory and interact favorably with the viscous terms.

For (G) and (H_{1},dots , H_{n}) finite groups, does there exist a 3-manifold group with (G) as a quotient but no (H_{i}) as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments.

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic (Phi ^{4}_{3})-model. This result is the hyperbolic counterpart to seminal works on the parabolic (Phi ^{4}_{3})-model by Hairer (Invent. Math. 198(2):269–504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651–1709, 2018).

The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates.

The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate.

We show that the mean curvature flow of generic closed surfaces in (mathbb{R}^{3}) avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in (mathbb{R}^{4}) is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least (C^{2}), we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class (r=2,ldots ,infty ,omega ). The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

We consider the distribution of the Galois groups (operatorname {Gal}(K^{operatorname{un}}/K)) of maximal unramified extensions as (K) ranges over (Gamma )-extensions of ℚ or ({{mathbb{F}}}_{q}(t)). We prove two properties of (operatorname {Gal}(K^{operatorname{un}}/K)) coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on (n)-generated profinite groups. In Part II, we prove as (qrightarrow infty ), agreement of (operatorname {Gal}(K^{operatorname{un}}/K)) as (K) varies over totally real (Gamma )-extensions of ({{mathbb{F}}}_{q}(t)) with our distribution from Part I, in the moments that are relatively prime to (q(q-1)|Gamma |). In particular, we prove for every finite group (Gamma ), in the (qrightarrow infty ) limit, the prime-to-(q(q-1)|Gamma |)-moments of the distribution of class groups of totally real (Gamma )-extensions of ({{mathbb{F}}}_{q}(t)) agree with the prediction of the Cohen–Lenstra–Martinet heuristics.