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European Mathematical Society Publishing House
2024-03-28 21:36:58
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=19&iss=11&update_since=2024-03-28
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
19
2017
11
McMullen polynomials and Lipschitz flows for free-by-cyclic groups
Spencer
Dowdall
University of Illinois at Urbana-Champaign, USA
Ilya
Kapovich
University of Illinois at Urbana-Champaign, USA
Christopher
Leininger
University of Illinois at Urbana-Champaign, USA
Free-by-cyclic groups, BNS-invariant, train track maps, stretch factors
Consider a group $G$ and an epimorphism $u_0\colon G\to \mathbb Z$ inducing a splitting of $G$ as a semidirect product ker$(u_0)\rtimes_\varphi \mathbb Z$ with ker$(u_0)$ a finitely generated free group and $\varphi\in$ Out (ker$(u_0)$) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized $G$ as $\pi_1(X)$ for an Eilenberg–Maclane 2-complex $X$ equipped with a semiflow $\psi$, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant $\mathfrak m \in \mathbb Z[H_1(G;\mathbb Z)/$torsion] for $(X,\psi)$ and investigate its properties. Specifically, $\mathfrak m$ determines a convex polyhedral cone $\mathcal C_X\subset H^1(G;\mathbb R)$, a convex, real-analytic function $\mathfrak H\colon \mathcal C_X\to \mathbb R$, and specializes to give an integral Laurent polynomial $\mathfrak m_u(\zeta)$ for each integral $u\in \mathcal C_X$. We show that $\mathcal C_X$ is equal to the "cone of sections" of $(X,\psi)$ (the convex hull of all cohomology classes dual to sections of of $\psi$), and that for each (compatible) cross section $\Theta_u\subset X$ with first return map $f_u\colon \Theta_u\to \Theta_u$, the specialization $\mathfrak m_u(\zeta)$ encodes the characteristic polynomial of the transition matrix of $f_u$. More generally, for every class $u\in \mathcal C_X$ there exists a geodesic metric $d_u$ and a codimension-1 foliation $\Omega_u$ of $X$ defined by a "closed 1-form" representing $u$ transverse to $\psi$ so that after reparametrizing the flow $\psi^u_{s}$ maps leaves of $\Omega_u$ to leaves via a local $e^{s\mathfrak H(u)}$-homothety. Among other things, we additionally prove that $\mathcal C_X$ is equal to (the cone over) the component of the BNS-invariant $\Sigma(G)$ containing $u_0$ and, consequently, that each primitive integral $u\in \mathcal C_X$ induces a splitting of $G$ as an ascending HNN-extension $G = Q_u\ast_{\phi_u}$ with $Q_u$ a finite-rank free group and $\phi_u\colon Q_u\to Q_u$ injective. For any such splitting, we show that the stretch factor of $\phi_u$ is exactly given by $e^{\mathfrak H(u)}$. In particular, we see that $\mathcal C_X$ and $\mathfrak H$ depend only on the group $G$ and epimorphism $u_0$.
Group theory and generalizations
Dynamical systems and ergodic theory
Manifolds and cell complexes
3253
3353
10.4171/JEMS/739
http://www.ems-ph.org/doi/10.4171/JEMS/739
A new isoperimetric inequality for the elasticae
Dorin
Bucur
Université de Savoie, Le-Bourget-du-Lac, France
Antoine
Henrot
Université de Lorraine, Vandoeuvre-lès-Nancy, France
Euler elasticae, minimization of elastic energy, isoperimetric inequality, curvature
For a smooth curve $\gamma$, we define its elastic energy as $E(\gamma)= \frac {1}{2} \int_{\gamma} k^2 (s) ds$ where $k(s)$ is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in $\mathbb R^2$, the disc has the boundary with the least elastic energy. In other words, for any bounded simply connected domain $\Omega$, the following isoperimetric inequality holds: $E^2(\partial \Omega)A(\Omega)\ge \pi ^3$. The analysis relies on the minimization of the elastic energy of drops enclosing a prescribed area, for which we give as well an analytic answer.
Calculus of variations and optimal control; optimization
Geometry
Differential geometry
3355
3376
10.4171/JEMS/740
http://www.ems-ph.org/doi/10.4171/JEMS/740
Conical structure for shrinking Ricci solitons
Ovidiu
Munteanu
University of Connecticut, Storrs, USA
Jiaping
Wang
University of Minnesota, Minneapolis, USA
Shrinking Ricci soliton, asymptotically conical
It is shown that a shrinking gradient Ricci soliton must be smoothly asymptotic to a cone if its Ricci curvature goes to zero at infinity.
Differential geometry
Global analysis, analysis on manifolds
3377
3390
10.4171/JEMS/741
http://www.ems-ph.org/doi/10.4171/JEMS/741
Algebraic embeddings of smooth almost complex structures
Jean-Pierre
Demailly
Université Grenoble Alpes, Gières, France
Hervé
Gaussier
Université Grenoble Alpes, Gières, France
Deformation of complex structures, almost complex manifolds, complex projective variety, Nijenhuis tensor, transverse embedding, Nash algebraic map
The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F. Bogomolov: can one embed every compact complex manifold as a $\mathcal C^\infty$ smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety?
Several complex variables and analytic spaces
Differential geometry
3391
3419
10.4171/JEMS/742
http://www.ems-ph.org/doi/10.4171/JEMS/742
The dynamical Manin–Mumford problem for plane polynomial automorphisms
Romain
Dujardin
Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France
Charles
Favre
École Polytechnique, Palaiseau, France
Dynamical Manin–Mumford problem, polynomial automorphisms of the plane, dynamical heights, arithmetic equidistribution, non-Archimedean dynamics, non-uniform hyperbolicity
Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian Jac $(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac $(f)$ is a root of unity. We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker–DeMarco and Yuan–Zhang.
Dynamical systems and ergodic theory
Several complex variables and analytic spaces
3421
3465
10.4171/JEMS/743
http://www.ems-ph.org/doi/10.4171/JEMS/743
Tame class field theory for singular varieties over finite fields
Thomas
Geisser
Rikkyo University, Tokyo, Japan
Alexander
Schmidt
Universität Heidelberg, Germany
Class field theory, Suslin homology, Weil-etale cohomology
Schmidt and Spieß described the abelian tame fundamental group of a smooth variety over a finite field by using Suslin homology. In this paper we show that their result generalizes to singular varieties if one uses Weil–Suslin homology instead.
$K$-theory
Number theory
Algebraic geometry
3467
3488
10.4171/JEMS/744
http://www.ems-ph.org/doi/10.4171/JEMS/744
Quantitative results on the corrector equation in stochastic homogenization
Antoine
Gloria
Université Libre de Bruxelles, Belgium and Team MEPHYSTO, Villeneuve d'Ascq, France
Felix
Otto
Max Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
Stochastic homogenization, corrector equation, variance estimate
We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d \geq 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb Z^d$ . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d > 2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages – the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.
Partial differential equations
Difference and functional equations
Probability theory and stochastic processes
3489
3548
10.4171/JEMS/745
http://www.ems-ph.org/doi/10.4171/JEMS/745
On sup-norms of cusp forms of powerful level
Abhishek
Saha
University of Bristol, UK
Maass form, sup-norm, Fourier coefficients, amplification
Let $f$ be an $L^2$-normalized Hecke–Maass cuspidal newform of level $N$ and Laplace eigenvalue $\lambda$. It is shown that $\|f\|_\infty \ll_{\lambda, \epsilon} N^{-1/12 + \epsilon}$ for any $\epsilon > 0$. The exponent is further improved in the case when $N$ is not divisible by "small squares". Our work extends and generalizes previously known results in the special case of $N$ squarefree.
Number theory
General
3549
3573
10.4171/JEMS/746
http://www.ems-ph.org/doi/10.4171/JEMS/746