Advances in Mathematics最新文献

{"title":"Two curious strongly invertible L-space knots","authors":"Kenneth L. Baker ,&nbsp;Marc Kegel ,&nbsp;Duncan McCoy","doi":"10.1016/j.aim.2025.110287","DOIUrl":"10.1016/j.aim.2025.110287","url":null,"abstract":"<div><div>We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110287"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869129","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":"On the dynamics of a complex continued fraction map which contains the Gauss map as its real number section","authors":"Hiromi Ei ,&nbsp;Hitoshi Nakada ,&nbsp;Rie Natsui","doi":"10.1016/j.aim.2025.110286","DOIUrl":"10.1016/j.aim.2025.110286","url":null,"abstract":"<div><div>We consider the complex continued fraction map <em>T</em> defined by R. Kaneiwa, I. Shiokawa, and J. Tamura (1975) associated with the Eisenstein field <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span>. A significant aspect of their continued fraction map is that the real number part of this map <em>T</em> is exactly the simple continued fraction map (Gauss map). In this paper we characterize the set of strictly periodic expansions of continued fraction expansions associated to this map in terms of quadratic extensions of <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span> in connection with the closure of <span><math><mo>{</mo><mo>−</mo><mfrac><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>, is the denominator of the <em>n</em>th principal convergent of the continued fraction expansion. Moreover, we show that the closure of <span><math><mo>{</mo><mo>−</mo><mfrac><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></math></span> has positive Lebesgue measure on the complex plane <span><math><mi>C</mi></math></span> though it has infinitely many holes. This gives us that the construction of the natural extension of <em>T</em> on a subset of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo><mo>{</mo><mtext>diagonal</mtext><mo>}</mo></math></span> is equivalent to the geodesics over the hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Then the invariant measure for the natural extension map is given by the hyperbolic measure. Hence its projection to the complex plane is obviously the invariant measure for <em>T</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110286"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870349","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":"Braid group actions on branched coverings and full exceptional sequences","authors":"Wen Chang ,&nbsp;Fabian Haiden ,&nbsp;Sibylle Schroll","doi":"10.1016/j.aim.2025.110284","DOIUrl":"10.1016/j.aim.2025.110284","url":null,"abstract":"<div><div>We relate full exceptional sequences in Fukaya categories of surfaces or equivalently in derived categories of graded gentle algebras to branched coverings over the disk, building on a previous classification result of the first and third author <span><span>[5]</span></span>. This allows us to apply tools from the theory of branched coverings such as Birman–Hilden theory and Hurwitz systems to study the natural braid group action on exceptional sequences. As an application, counterexamples are given to a conjecture of Bondal–Polishchuk <span><span>[3]</span></span> on the transitivity of the braid group action on full exceptional sequences in a triangulated category.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110284"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870348","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":"Large-scale geometry of Borel graphs of polynomial growth","authors":"Anton Bernshteyn ,&nbsp;Jing Yu","doi":"10.1016/j.aim.2025.110290","DOIUrl":"10.1016/j.aim.2025.110290","url":null,"abstract":"<div><div>We study graphs of polynomial growth from the perspective of asymptotic geometry and descriptive set theory. The starting point of our investigation is a theorem of Krauthgamer and Lee who showed that every connected graph of polynomial growth admits an injective contraction mapping to <span><math><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mo>‖</mo><mo>⋅</mo><mo>‖</mo></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span> for some <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. We strengthen and generalize this result in a number of ways. In particular, answering a question of Papasoglu, we construct coarse embeddings from graphs of polynomial growth to <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Moreover, we only require <em>n</em> to be linear in the asymptotic polynomial growth rate of the graph; this confirms a conjecture of Levin and Linial, London, and Rabinovich “in the asymptotic sense.” (The exact form of the conjecture was refuted by Krauthgamer and Lee.) All our results are proved for Borel graphs, which allows us to settle a number of problems in descriptive combinatorics. Roughly, we prove that graphs generated by free Borel actions of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> are universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>-actions to all Borel graphs of polynomial growth. For example, an immediate consequence of our main result is that all Borel graphs of polynomial growth are hyperfinite, which answers a well-known question in the area. As another illustration, we show that Borel graphs of polynomial growth support a certain combinatorial structure called a toast. An important technical tool in our arguments is the notion of padded decomposition from computer science, which is closely related to the concept of asymptotic dimension due to Gromov. Along the way we find an alternative, probabilistic proof of a theorem of Papasoglu that graphs of asymptotic polynomial growth rate <span><math><mi>ρ</mi><mo>&lt;</mo><mo>∞</mo></math></span> have asymptotic dimension at most <em>ρ</em> and establish the same bound in the Borel setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110290"},"PeriodicalIF":1.5,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143863932","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":"2-torsion in instanton Floer homology","authors":"Zhenkun Li ,&nbsp;Fan Ye","doi":"10.1016/j.aim.2025.110289","DOIUrl":"10.1016/j.aim.2025.110289","url":null,"abstract":"<div><div>This paper studies the existence of 2-torsion in instanton Floer homology with <span><math><mi>Z</mi></math></span> coefficients for closed 3-manifolds and singular knots. First, we show that the non-existence of 2-torsion in the framed instanton Floer homology <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>♯</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo><mo>;</mo><mi>Z</mi><mo>)</mo></math></span> of any nonzero integral <em>n</em>-surgery along a knot <em>K</em> in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> would imply that <em>K</em> is fibered. Also, we show that <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>♯</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo><mo>;</mo><mi>Z</mi><mo>)</mo></math></span> for any nontrivial <em>K</em> with <span><math><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>4</mn></math></span> always has 2-torsion. These two results indicate that the existence of 2-torsion is expected to be a generic phenomenon for Dehn surgeries along knots. Second, we show that for genus-one knots with nontrivial Alexander polynomials and for unknotting-number-one knots, the unreduced singular instanton knot homology <span><math><msup><mrow><mi>I</mi></mrow><mrow><mo>♯</mo></mrow></msup><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mi>K</mi><mo>;</mo><mi>Z</mi><mo>)</mo></math></span> always has 2-torsion. Finally, some crucial lemmas that help us demonstrate the existence of 2-torsion are motivated by analogous results in Heegaard Floer theory, which may be of independent interest. In particular, we show that, for a knot <em>K</em> in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, if there is a nonzero rational number <em>r</em> such that the dual knot <span><math><msub><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msub></math></span> inside <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo></math></span> is Floer simple, then <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo></math></span> must be an L-space and <em>K</em> must be an L-space knot.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110289"},"PeriodicalIF":1.5,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864896","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":"Restricted Hausdorff spectra of q-adic automorphisms","authors":"Jorge Fariña-Asategui","doi":"10.1016/j.aim.2025.110294","DOIUrl":"10.1016/j.aim.2025.110294","url":null,"abstract":"<div><div>Firstly, we completely determine the self-similar Hausdorff spectrum of the group of <em>q</em>-adic automorphisms where <em>q</em> is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products.</div><div>Secondly, we provide for every prime <em>p</em> the first examples of just infinite branch pro-<em>p</em> groups with zero Hausdorff dimension in <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, giving strong evidence against a well-known conjecture of Boston.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110294"},"PeriodicalIF":1.5,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864897","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":"Frieze patterns and Farey complexes","authors":"Ian Short,&nbsp;Matty van Son,&nbsp;Andrei Zabolotskii","doi":"10.1016/j.aim.2025.110269","DOIUrl":"10.1016/j.aim.2025.110269","url":null,"abstract":"<div><div>Frieze patterns have attracted significant attention recently, motivated by their relationship with cluster algebras. A longstanding open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo <em>N</em> akin to Conway and Coxeter's celebrated model for positive integer frieze patterns. Here we solve this problem using the Farey complex of the ring of integers modulo <em>N</em>; in fact, using more general Farey complexes we provide combinatorial models for frieze patterns over any rings whatsoever.</div><div>Our strategy generalises that of the first author and of Morier-Genoud et al. for integers and that of Felikson et al. for Eisenstein integers. We also generalise results of Singerman and Strudwick on diameters of Farey graphs, we recover a theorem of Morier-Genoud on enumerating friezes over finite fields, and we classify those frieze patterns modulo <em>N</em> that lift to frieze patterns over the integers in terms of the topology of the corresponding Farey complexes.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110269"},"PeriodicalIF":1.5,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859856","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":"Codimension two polar homogeneous foliations on symmetric spaces of noncompact type","authors":"José Carlos Díaz-Ramos,&nbsp;Juan Manuel Lorenzo-Naveiro","doi":"10.1016/j.aim.2025.110295","DOIUrl":"10.1016/j.aim.2025.110295","url":null,"abstract":"<div><div>We classify homogeneous polar foliations of codimension two on irreducible symmetric spaces of noncompact type up to orbit equivalence. Any such foliation is either hyperpolar or the canonical extension of a polar homogeneous foliation on a rank one symmetric space.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"472 ","pages":"Article 110295"},"PeriodicalIF":1.5,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859857","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":"Erratum to “Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes” [Adv. Math. 415 (2023) 108890]","authors":"Ugo Bruzzo ,&nbsp;Daniel Hernández Ruipérez ,&nbsp;Alexander Polishchuk","doi":"10.1016/j.aim.2025.110273","DOIUrl":"10.1016/j.aim.2025.110273","url":null,"abstract":"","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110273"},"PeriodicalIF":1.5,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143833473","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 inverse limit topology and profinite descent on Picard groups in K(n)-local homotopy theory","authors":"Guchuan Li ,&nbsp;Ningchuan Zhang","doi":"10.1016/j.aim.2025.110274","DOIUrl":"10.1016/j.aim.2025.110274","url":null,"abstract":"<div><div>In this paper, we study profinite descent theory for Picard groups in <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-local homotopy theory through their inverse limit topology. Building upon Burklund's result on the multiplicative structures of generalized Moore spectra, we prove that the module category over a <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-local commutative ring spectrum is equivalent to the limit of its base changes by a tower of generalized Moore spectra of type <em>n</em>. As a result, the <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-local Picard groups are endowed with a natural inverse limit topology. This topology allows us to identify the entire <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-pages of a descent spectral sequence for Picard spaces of <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-local profinite Galois extensions.</div><div>Our main examples are <span><math><mi>K</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-local Picard groups of homotopy fixed points <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi><mi>G</mi></mrow></msubsup></math></span> of the Morava <em>E</em>-theory <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all closed subgroups <em>G</em> of the Morava stabilizer group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> case has been studied by Heard and Mor. At height 1, we compute Picard groups of <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>h</mi><mi>G</mi></mrow></msubsup></math></span> for all closed subgroups <em>G</em> of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span> at all primes as a Mackey functor.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"471 ","pages":"Article 110274"},"PeriodicalIF":1.5,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143828302","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}