We prove that every Besicovitch set in R3mathbb {R}^3 must have Hausdorff dimension at least 5/2+ϵ05/2+epsilon _0 for some small constant ϵ0>0epsilon _0>0. This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the SL2operatorname {SL}_2 example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension 5/25/2. We believe this example may be an interesting object for future study.
{"title":"An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³","authors":"N. Katz, Joshua Zahl","doi":"10.1090/jams/907","DOIUrl":"https://doi.org/10.1090/jams/907","url":null,"abstract":"<p>We prove that every Besicovitch set in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R cubed\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> must have Hausdorff dimension at least <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"5 slash 2 plus epsilon 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>5</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>ϵ<!-- ϵ --></mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">5/2+epsilon _0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for some small constant <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon 0 greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>ϵ<!-- ϵ --></mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">epsilon _0>0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S upper L 2\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>SL</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">operatorname {SL}_2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"5 slash 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>5</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">5/2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We believe this example may be an interesting object for future study.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/907","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41674663","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}
Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) pi _2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.
{"title":"Homological mirror symmetry without correction","authors":"M. Abouzaid","doi":"10.1090/JAMS/973","DOIUrl":"https://doi.org/10.1090/JAMS/973","url":null,"abstract":"Let \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base \u0000\u0000 \u0000 Q\u0000 Q\u0000 \u0000\u0000. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space \u0000\u0000 \u0000 Y\u0000 Y\u0000 \u0000\u0000, which can be considered as a variant of the \u0000\u0000 \u0000 T\u0000 T\u0000 \u0000\u0000-dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 embeds fully faithfully in the derived category of (twisted) coherent sheaves on \u0000\u0000 \u0000 Y\u0000 Y\u0000 \u0000\u0000, under the technical assumption that \u0000\u0000 \u0000 \u0000 \u0000 π\u0000 2\u0000 \u0000 (\u0000 Q\u0000 )\u0000 \u0000 pi _2(Q)\u0000 \u0000\u0000 vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46586983","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}
In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.
{"title":"Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters","authors":"Dzmitry Dudko, M. Lyubich, N. Selinger","doi":"10.1090/jams/942","DOIUrl":"https://doi.org/10.1090/jams/942","url":null,"abstract":"In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of \"Pacman Renormalization Theory\" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/942","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43945566","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}
We show that every quasisymmetric homeomorphism of the circle ∂H^2 admits a harmonic quasiconformal extension to the hyperbolic plane H^2. This proves the Schoen conjecture.
{"title":"Harmonic maps and the Schoen conjecture","authors":"V. Marković","doi":"10.1090/JAMS/881","DOIUrl":"https://doi.org/10.1090/JAMS/881","url":null,"abstract":"We show that every quasisymmetric homeomorphism of the circle ∂H^2 admits a harmonic quasiconformal extension to the hyperbolic plane H^2. This proves the Schoen conjecture.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"30 1","pages":"799-817"},"PeriodicalIF":3.9,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/881","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44942672","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}
A. Chambolle, M. Morini, M. Novaga, M. Ponsiglione
An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor, and Wang (1993) to a unique (up to fattening) “flat flow” in the case of general, including crystalline, anisotropies, solving a long-standing open question.
{"title":"Existence and uniqueness for anisotropic and crystalline mean curvature flows","authors":"A. Chambolle, M. Morini, M. Novaga, M. Ponsiglione","doi":"10.1090/JAMS/919","DOIUrl":"https://doi.org/10.1090/JAMS/919","url":null,"abstract":"An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor, and Wang (1993) to a unique (up to fattening) “flat flow” in the case of general, including crystalline, anisotropies, solving a long-standing open question.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/919","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45851638","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}
We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.
{"title":"Mod points on Shimura varieties of abelian type","authors":"M. Kisin","doi":"10.1090/JAMS/867","DOIUrl":"https://doi.org/10.1090/JAMS/867","url":null,"abstract":"We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"30 1","pages":"819-914"},"PeriodicalIF":3.9,"publicationDate":"2017-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/867","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47959055","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}
M. Bhargava, A. Shankar, Takashi Taniguchi, F. Thorne, Jacob Tsimerman, Yongqiang Zhao
<p>We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the trivial bound being <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O Subscript epsilon comma n Baseline left-parenthesis StartAbsoluteValue normal upper D normal i normal s normal c left-parenthesis upper K right-parenthesis EndAbsoluteValue Superscript 1 slash 2 plus epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">D</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ<!-- ϵ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O_{epsilon ,n}(|mathrm {Disc}(K)|^{1/2+epsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, (3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves, and (4) bounds of Baily and Wong on the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 4"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">A_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-quartic fields of bound
我们证明了三次和高次数域K K的类群的2-扭子群的大小的第一个已知的非平凡界（平凡界是O∈，n（|D i s c（K）|1/2+∈）O_｛epsilon，n｝（|mathrm｛Disc｝（K）|^{1/2+epsilon}）。这得到了相应的改进：（1）Brumer和Kramer关于2-Selmer群的大小和椭圆曲线的秩的界，（2）Helfgott和Venkatesh关于椭圆曲线上积分点数的界，（4）Baily和Wong关于有界判别式的A4 A_ 4-四次域个数的界。
{"title":"Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves","authors":"M. Bhargava, A. Shankar, Takashi Taniguchi, F. Thorne, Jacob Tsimerman, Yongqiang Zhao","doi":"10.1090/jams/945","DOIUrl":"https://doi.org/10.1090/jams/945","url":null,"abstract":"<p>We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (the trivial bound being <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O Subscript epsilon comma n Baseline left-parenthesis StartAbsoluteValue normal upper D normal i normal s normal c left-parenthesis upper K right-parenthesis EndAbsoluteValue Superscript 1 slash 2 plus epsilon Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>O</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>ϵ<!-- ϵ --></mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">D</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">i</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">s</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">c</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>ϵ<!-- ϵ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">O_{epsilon ,n}(|mathrm {Disc}(K)|^{1/2+epsilon })</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, (3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves, and (4) bounds of Baily and Wong on the number of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A 4\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">A_4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-quartic fields of bound","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/945","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43792237","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}
A well-known question of Gromov asks whether every one-ended hyperbolic group Γ Gamma has a surface subgroup. We give a positive answer when Γ Gamma is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov’s question is reduced (modulo a technical assumption on 2-torsion) to the case when Γ Gamma is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.
{"title":"Essential surfaces in graph pairs","authors":"H. Wilton","doi":"10.1090/JAMS/901","DOIUrl":"https://doi.org/10.1090/JAMS/901","url":null,"abstract":"A well-known question of Gromov asks whether every one-ended hyperbolic group \u0000\u0000 \u0000 Γ\u0000 Gamma\u0000 \u0000\u0000 has a surface subgroup. We give a positive answer when \u0000\u0000 \u0000 Γ\u0000 Gamma\u0000 \u0000\u0000 is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov’s question is reduced (modulo a technical assumption on 2-torsion) to the case when \u0000\u0000 \u0000 Γ\u0000 Gamma\u0000 \u0000\u0000 is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":3.9,"publicationDate":"2017-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/901","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41580566","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":"Simple groups of Morley rank 3 are algebraic","authors":"Olivier Frécon","doi":"10.1090/JAMS/892","DOIUrl":"https://doi.org/10.1090/JAMS/892","url":null,"abstract":"","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"31 1","pages":"643-659"},"PeriodicalIF":3.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/892","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551995","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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