{"title":"Direct Products of Free Groups in Aut(FN)","authors":"Martin R. Bridson, Richard D. Wade","doi":"10.1007/s00039-024-00688-5","DOIUrl":"https://doi.org/10.1007/s00039-024-00688-5","url":null,"abstract":"<p>We give a complete description of the embeddings of direct products of nonabelian free groups into Aut(<i>F</i><sub><i>N</i></sub>) and Out(<i>F</i><sub><i>N</i></sub>) when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141891851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Maximal Multiplicity of Laplacian Eigenvalues in Negatively Curved Surfaces","authors":"Cyril Letrouit, Simon Machado","doi":"10.1007/s00039-024-00691-w","DOIUrl":"https://doi.org/10.1007/s00039-024-00691-w","url":null,"abstract":"<p>In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus <i>g</i>. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an <i>r</i>-net in the surface to control this trace. This last idea was introduced in 2021 for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the “approximate multiplicity” of eigenvalues, i.e., the number of eigenvalues in windows of size 1/log<sup><i>β</i></sup>(<i>g</i>), <i>β</i>>0. This work provides new insights on a conjecture by Colin de Verdière and new ways to transfer spectral results from graphs to surfaces.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141768457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mass Equidistribution for Saito-Kurokawa Lifts","authors":"Jesse Jääsaari, Stephen Lester, Abhishek Saha","doi":"10.1007/s00039-024-00690-x","DOIUrl":"https://doi.org/10.1007/s00039-024-00690-x","url":null,"abstract":"<p>Let <i>F</i> be a holomorphic cuspidal Hecke eigenform for <span>(mathrm{Sp}_{4}({mathbb{Z}}))</span> of weight <i>k</i> that is a Saito–Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of <i>F</i> equidistributes on the Siegel modular variety as <i>k</i>⟶∞. As a corollary, we show under GRH that the zero divisors of Saito–Kurokawa lifts equidistribute as their weights tend to infinity.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Virtually Free-by-Cyclic Groups","authors":"Dawid Kielak, Marco Linton","doi":"10.1007/s00039-024-00687-6","DOIUrl":"https://doi.org/10.1007/s00039-024-00687-6","url":null,"abstract":"<p>We obtain a homological characterisation of virtually free-by-cyclic groups among groups that are hyperbolic and virtually compact special. As a consequence, we show that many groups known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, we show that all one-relator groups with torsion are virtually free-by-cyclic, solving a conjecture of Baumslag.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Disk-Like Surfaces of Section and Symplectic Capacities","authors":"O. Edtmair","doi":"10.1007/s00039-024-00689-4","DOIUrl":"https://doi.org/10.1007/s00039-024-00689-4","url":null,"abstract":"<p>We prove that the cylindrical capacity of a dynamically convex domain in <span>({mathbb{R}}^{4})</span> agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in <span>({mathbb{R}}^{4})</span> which are sufficiently <i>C</i><sup>3</sup> close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141631497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Singular Support of Sheaves Is γ-Coisotropic","authors":"Stéphane Guillermou, Claude Viterbo","doi":"10.1007/s00039-024-00682-x","DOIUrl":"https://doi.org/10.1007/s00039-024-00682-x","url":null,"abstract":"<p>We prove that the singular support of an element in the derived category of sheaves is <i>γ</i>-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense of Kashiwara-Schapira, but being <i>γ</i>-coisotropic has the advantage to be invariant by symplectic homeomorphisms (while involutivity is only invariant by <i>C</i><sup>1</sup> diffeomorphisms) and we give an example of an involutive set that is not <i>γ</i>-coisotropic. Along the way we prove a number of results relating the singular support and the spectral norm <i>γ</i> and raise a number of new questions.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fusion and Positivity in Chiral Conformal Field Theory","authors":"James E. Tener","doi":"10.1007/s00039-024-00685-8","DOIUrl":"https://doi.org/10.1007/s00039-024-00685-8","url":null,"abstract":"<p>In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann’s landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. ‘fusion’) of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of ‘bounded localized vertex operators,’ which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141461902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Growth of k-Dimensional Systoles in Congruence Coverings","authors":"Mikhail Belolipetsky, Shmuel Weinberger","doi":"10.1007/s00039-024-00686-7","DOIUrl":"https://doi.org/10.1007/s00039-024-00686-7","url":null,"abstract":"<p>We study growth of absolute and homological <i>k</i>-dimensional systoles of arithmetic <i>n</i>-manifolds along congruence coverings. Our main interest is in the growth of systoles of manifolds whose real rank <i>r</i>≥2. We observe, in particular, that in some cases for <i>k</i>=<i>r</i> the growth function tends to oscillate between a power of a logarithm and a power function of the degree of the covering. This is a new phenomenon. We also prove the expected polylogarithmic and constant power bounds for small and large <i>k</i>, respectively.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rigidity Theorems for Higher Rank Lattice Actions","authors":"Homin Lee","doi":"10.1007/s00039-024-00683-w","DOIUrl":"https://doi.org/10.1007/s00039-024-00683-w","url":null,"abstract":"<p>Let Γ be a weakly irreducible lattice in a higher rank semisimple algebraic Lie group <i>G</i> without property (T) such as <span>(mathrm {SL}_{2}({mathbb{Z}}[sqrt{2}]))</span> in <span>(mathrm {SL}_{2}({mathbb{R}})times mathrm {SL}_{2}({mathbb{R}}))</span>.</p><p>In this paper, for such Γ, we prove a cocycle superrigidity, <i>Dynamical cocycle superrigidity</i>, for Γ-actions under <i>L</i><sup>2</sup> integrability and irreducibility assumptions. It gives a similar result to Zimmer’s cocycle superrigidity, so we can apply it instead of Zimmer’s cocycle superrigidity in many situations. For instance, in this paper, we obtain global rigidity of Anosov Γ-actions on nilmanifolds under the irreducibility assumption on a fully supported invariant measure.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows","authors":"Davide Parise, Alessandro Pigati, Daniel Stern","doi":"10.1007/s00039-024-00684-9","DOIUrl":"https://doi.org/10.1007/s00039-024-00684-9","url":null,"abstract":"<p>We develop the asymptotic analysis as <i>ε</i>→0 for the natural gradient flow of the self-dual <i>U</i>(1)-Higgs energies </p><span>$$ E_{varepsilon }(u,nabla )=int _{M}left (|nabla u|^{2}+ varepsilon ^{2}|F_{nabla }|^{2}+ frac{(1-|u|^{2})^{2}}{4varepsilon ^{2}}right ) $$</span><p> on Hermitian line bundles over closed manifolds (<i>M</i><sup><i>n</i></sup>,<i>g</i>) of dimension <i>n</i>≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (<i>n</i>−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (<i>n</i>−2)-cycle Γ<sub>0</sub> in <i>M</i>, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual <i>U</i>(1)-Yang–Mills–Higgs energies to the (<i>n</i>−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ<sub>0</sub> with additional structure, similar to those produced via Ilmanen’s elliptic regularization.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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