James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson, Nik Ruškuc
{"title":"On the diameter of semigroups of transformations and partitions","authors":"James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson, Nik Ruškuc","doi":"10.1112/jlms.12944","DOIUrl":"https://doi.org/10.1112/jlms.12944","url":null,"abstract":"<p>For a semigroup <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$FP_1$</annotation>\u0000 </semantics></math>), the right diameter of <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> is a parameter that expresses how ‘far apart’ elements of <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> can be from each other, in a certain sense. To be more precise, for each finite generating set <span></span><math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$U$</annotation>\u0000 </semantics></math> for the universal right congruence on <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>, we have a metric space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>d</mi>\u0000 <mi>U</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(S,d_U)$</annotation>\u0000 </semantics></math> where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>d</mi>\u0000 <mi>U</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$d_U(a,b)$</annotation>\u0000 </semantics></math> is the minimum length of derivations for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(a,b)$</annotation>\u0000 </semantics></math> as a consequence of pairs in <span></span><math>\u0000 <semantics>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12944","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141326634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Relatively Anosov representations via flows II: Examples","authors":"Feng Zhu, Andrew Zimmer","doi":"10.1112/jlms.12949","DOIUrl":"https://doi.org/10.1112/jlms.12949","url":null,"abstract":"<p>This is the second in a series of two papers that develops a theory of relatively Anosov representations using the original “contracting flow on a bundle” definition of Anosov representations introduced by Labourie and Guichard–Wienhard. In this paper, we focus on building families of examples.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12949","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin
{"title":"Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type","authors":"Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin","doi":"10.1112/jlms.12946","DOIUrl":"https://doi.org/10.1112/jlms.12946","url":null,"abstract":"<p>We study the <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℏ</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$hbar ^2$</annotation>\u0000 </semantics></math>-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so-called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ℏ</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$hbar ^2$</annotation>\u0000 </semantics></math>-deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)-type formulas discussed in the literature.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12946","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Characterizing slopes for \u0000 \u0000 \u0000 5\u0000 2\u0000 \u0000 $5_2$","authors":"John A. Baldwin, Steven Sivek","doi":"10.1112/jlms.12951","DOIUrl":"https://doi.org/10.1112/jlms.12951","url":null,"abstract":"<p>We prove that all rational slopes are characterizing for the knot <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mn>5</mn>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$5_2$</annotation>\u0000 </semantics></math>, except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$S^3$</annotation>\u0000 </semantics></math> that produce the Brieskorn sphere <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Σ</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 <mo>,</mo>\u0000 <mn>11</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Sigma (2,3,11)$</annotation>\u0000 </semantics></math>, and we study knots on which large integral surgeries are almost L-spaces.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}