{"title":"The maximal subgroups of the Monster","authors":"Heiko Dietrich, Melissa Lee, Tomasz Popiel","doi":"10.1016/j.aim.2025.110214","DOIUrl":"10.1016/j.aim.2025.110214","url":null,"abstract":"<div><div>The classification of the maximal subgroups of the Monster <strong>M</strong> is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether <strong>M</strong> contains maximal subgroups that are almost simple with socle <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>13</mn><mo>)</mo></math></span>. However, this conclusion relies on reported claims, with unpublished proofs, that <strong>M</strong> has no maximal subgroups that are almost simple with socle <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>8</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>16</mn><mo>)</mo></math></span>, or <span><math><msub><mrow><mi>PSU</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mn>4</mn><mo>)</mo></math></span>. The aim of this paper is to settle all of these questions, and thereby complete the solution to the maximal subgroup problem for <strong>M</strong>, and for the sporadic simple groups as a whole. Specifically, we prove the existence of two new maximal subgroups of <strong>M</strong>, isomorphic to the automorphism groups of <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>13</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>PSU</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mn>4</mn><mo>)</mo></math></span>, and we establish that <strong>M</strong> has no almost simple maximal subgroup with socle <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>8</mn><mo>)</mo></math></span> or <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>16</mn><mo>)</mo></math></span>. We also correct the claim that <strong>M</strong> has no almost simple maximal subgroup with socle <span><math><msub><mrow><mi>PSU</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mn>4</mn><mo>)</mo></math></span>, and provide evidence that the maximal subgroup <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>59</mn><mo>)</mo></math></span> (constructed in 2004) does not exist. Our proofs are supported by reproducible computations carried out using the publicly available Python package <span>mmgroup</span> for computing with <strong>M</strong> recently developed by M. Seysen. We provide explicit generators for our newly discovered maximal subgroups of <strong>M</strong> in <span>mmgroup</span> format.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110214"},"PeriodicalIF":1.5,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143705004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vasileios Chousionis , Katrin Fässler , Tuomas Orponen
{"title":"Corrigendum to: “Boundedness of singular integrals on C1,α intrinsic graphs in the Heisenberg group”","authors":"Vasileios Chousionis , Katrin Fässler , Tuomas Orponen","doi":"10.1016/j.aim.2025.110224","DOIUrl":"10.1016/j.aim.2025.110224","url":null,"abstract":"","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110224"},"PeriodicalIF":1.5,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Farkhod Eshmatov , Xabier García-Martínez , Rustam Turdibaev
{"title":"Noncommutative Poisson structure and invariants of matrices","authors":"Farkhod Eshmatov , Xabier García-Martínez , Rustam Turdibaev","doi":"10.1016/j.aim.2025.110212","DOIUrl":"10.1016/j.aim.2025.110212","url":null,"abstract":"<div><div>We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices. We entirely solve the open problem of computing the algebra of invariants of two <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> matrices. As an application, we derive the complete description of the invariant commuting variety of pairs of <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> matrices and the fourth Calogero-Moser space.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110212"},"PeriodicalIF":1.5,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679780","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":"Iterated magnitude homology","authors":"Emily Roff","doi":"10.1016/j.aim.2025.110210","DOIUrl":"10.1016/j.aim.2025.110210","url":null,"abstract":"<div><div>Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology—the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction: it extends from categories to bicategories as the geometric realization of the geometric nerve. This paper introduces a hybrid of the two ideas: an <em>iterated magnitude homology</em> theory for categories with a second- or higher-order enrichment. This encompasses, for example, groups equipped with extra structure such as a partial ordering or a bi-invariant metric. In the case of a strict 2-category, iterated magnitude homology recovers the homology of the classifying space; we investigate its content and behaviour when interpreted for partially ordered groups, normed groups, and strict <em>n</em>-categories for <span><math><mi>n</mi><mo>></mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110210"},"PeriodicalIF":1.5,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simultaneous Diophantine approximation to points on the Veronese curve","authors":"Dmitry Badziahin","doi":"10.1016/j.aim.2025.110213","DOIUrl":"10.1016/j.aim.2025.110213","url":null,"abstract":"<div><div>We compute the Hausdorff dimension of the set of simultaneously <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mi>λ</mi></mrow></msup></math></span>-well approximable points on the Veronese curve in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for <em>λ</em> between <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span> and <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span>. For <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>, the same result is given for a wider range of <em>λ</em> between <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. We also provide a nontrivial upper bound for this Hausdorff dimension in the case <span><math><mi>λ</mi><mo>⩽</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>. In the course of the proof we establish that the number of cubic polynomials of height at most <em>H</em> and non-zero discriminant at most <em>D</em> is bounded from above by <span><math><mi>c</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>ϵ</mi></mrow></msup><msup><mrow><mi>D</mi></mrow><mrow><mn>5</mn><mo>/</mo><mn>6</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110213"},"PeriodicalIF":1.5,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Monodromy of generalized Lamé equations with Darboux-Treibich-Verdier potentials: A universal law","authors":"Zhijie Chen , Chang-Shou Lin","doi":"10.1016/j.aim.2025.110209","DOIUrl":"10.1016/j.aim.2025.110209","url":null,"abstract":"<div><div>The Darboux-Treibich-Verdier (DTV) potential <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mn>3</mn></mrow></msubsup><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><mn>1</mn><mo>)</mo><mo>℘</mo><mo>(</mo><mi>z</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><mi>τ</mi><mo>)</mo></math></span> is well-known as a (parametric) doubly-periodic solution of the stationary KdV hierarchy (Treibich and Verdier, 1992) <span><span>[42]</span></span>. In this paper, we study the generalized Lamé equation with the DTV potential<span><span><span><math><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mo>[</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mn>3</mn></mrow></munderover><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><mn>1</mn><mo>)</mo><mo>℘</mo><mo>(</mo><mi>z</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><mi>τ</mi><mo>)</mo><mo>+</mo><mi>B</mi><mo>]</mo><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi>N</mi></math></span></span></span> from the (doubly periodic) monodromy aspect. This equation is the elliptic form of the well-known integral Heun equation. We prove that the map from <span><math><mo>(</mo><mi>τ</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> to the monodromy data <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> satisfies a surprising universal law <span><math><mi>d</mi><mi>τ</mi><mo>∧</mo><mi>d</mi><mi>B</mi><mo>≡</mo><mn>8</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>r</mi><mo>∧</mo><mi>d</mi><mi>s</mi></math></span>. Our proof applies the Panlevé VI equation and modular forms. We also give applications to compute the algebraic multiplicity of the (anti)periodic eigenvalues for the associated Hill operator.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110209"},"PeriodicalIF":1.5,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683757","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}
Anna Beliakova , Matthew Hogancamp , Krzysztof Putyra , Stephan Wehrli
{"title":"On unification of colored annular sl2 knot homology","authors":"Anna Beliakova , Matthew Hogancamp , Krzysztof Putyra , Stephan Wehrli","doi":"10.1016/j.aim.2025.110206","DOIUrl":"10.1016/j.aim.2025.110206","url":null,"abstract":"<div><div>We show that the Khovanov and Cooper–Krushkal models for colored <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> homology are equivalent in the case of the unknot, when formulated in the quantum annular Bar-Natan category. Again for the unknot, these two theories are shown to be equivalent to a third colored homology theory, defined using the action of Jones–Wenzl projectors on the quantum annular homology of cables. The proof is given by conceptualizing the properties of all three models into a Chebyshev system and by proving its uniqueness. In addition, we show that the classes of the Cooper–Hogancamp projectors in the quantum horizontal trace coincide with those of the Cooper–Krushkal projectors on the passing through strands. As an application, we compute the full quantum Hochschild homology of Khovanov's arc algebras. Finally, we state precise conjectures formalizing cabling operations and extending the above results to all knots.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110206"},"PeriodicalIF":1.5,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643591","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}
Yann Bugeaud , Gerardo González Robert , Mumtaz Hussain
{"title":"Metrical properties of Hurwitz continued fractions","authors":"Yann Bugeaud , Gerardo González Robert , Mumtaz Hussain","doi":"10.1016/j.aim.2025.110208","DOIUrl":"10.1016/j.aim.2025.110208","url":null,"abstract":"<div><div>We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.</div><div>Let <span><math><mi>Φ</mi><mo>:</mo><mi>N</mi><mo>→</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span> be any function. For any complex number <em>z</em> and <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>, let <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span> denote the <em>n</em>th partial quotient in the Hurwitz continued fraction of <em>z</em>. One of the main results of this paper is the computation of the Hausdorff dimension of the set<span><span><span><math><mi>E</mi><mo>(</mo><mi>Φ</mi><mo>)</mo><mo>:</mo><mo>=</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mi>Φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mtext> for infinitely many </mtext><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></mrow><mo>.</mo></math></span></span></span> This study is a complex analog of a well-known result of Wang and Wu (2008) <span><span>[55]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110208"},"PeriodicalIF":1.5,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms","authors":"Bram Mesland , Adam Rennie","doi":"10.1016/j.aim.2025.110207","DOIUrl":"10.1016/j.aim.2025.110207","url":null,"abstract":"<div><div>We combine Hilbert module and algebraic techniques to give necessary and sufficient conditions for the existence of an Hermitian torsion-free connection on the bimodule of differential one-forms of a first order differential calculus. In the presence of the extra structure of a bimodule connection, we give sufficient conditions for uniqueness.</div><div>We prove that any <em>θ</em>-deformation of a compact Riemannian manifold admits a unique Hermitian torsion-free bimodule connection and provide an explicit construction of it. Specialising to classical Riemannian manifolds yields a novel construction of the Levi-Civita connection on the cotangent bundle.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110207"},"PeriodicalIF":1.5,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups","authors":"Tullia Dymarz , David Fisher , Xiangdong Xie","doi":"10.1016/j.aim.2025.110202","DOIUrl":"10.1016/j.aim.2025.110202","url":null,"abstract":"<div><div>In this paper we study uniform quasiconformal groups of Carnot-by-Carnot groups. We show that they can be conjugated into conformal groups provided the induced action on the space of distinct pairs is cocompact. Following the approach of Eskin-Fisher-Whyte <span><span>[17]</span></span> these results have applications to quasi-isometric rigidity of certain families of solvable groups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"468 ","pages":"Article 110202"},"PeriodicalIF":1.5,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636832","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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