Mathematical Proceedings of the Cambridge Philosophical Society最新文献

{"title":"Generalised knotoids","authors":"COLIN ADAMS, ZACHARY ROMRELL, ALEXANDRA BONAT, MAYA CHANDE, JOYE CHEN, MAXWELL JIANG, DANIEL SANTIAGO, BENJAMIN SHAPIRO, DORA WOODRUFF","doi":"10.1017/s0305004124000148","DOIUrl":"https://doi.org/10.1017/s0305004124000148","url":null,"abstract":"<p>In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Domination inequalities and dominating graphs","authors":"DAVID CONLON, JOONKYUNG LEE","doi":"10.1017/s0305004124000185","DOIUrl":"https://doi.org/10.1017/s0305004124000185","url":null,"abstract":"We say that a graph <jats:italic>H</jats:italic> dominates another graph <jats:italic>H</jats:italic><jats:sup>′</jats:sup> if the number of homomorphisms from <jats:italic>H</jats:italic><jats:sup>′</jats:sup> to any graph <jats:italic>G</jats:italic> is dominated, in an appropriate sense, by the number of homomorphisms from <jats:italic>H</jats:italic> to <jats:italic>G</jats:italic>. We study the family of dominating graphs, those graphs with the property that they dominate all of their subgraphs. It has long been known that even-length paths are dominating in this sense and a result of Hatami implies that all weakly norming graphs are dominating. In a previous paper, we showed that every finite reflection group gives rise to a family of weakly norming, and hence dominating, graphs. Here we revisit this connection to show that there is a much broader class of dominating graphs.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"3 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiplicative dependence of rational values modulo approximate finitely generated groups","authors":"ATTILA BÉRCZES, YANN BUGEAUD, KÁLMÁN GYŐRY, JORGE MELLO, ALINA OSTAFE, MIN SHA","doi":"10.1017/s0305004124000173","DOIUrl":"https://doi.org/10.1017/s0305004124000173","url":null,"abstract":"<p>In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field <span>K</span>. For example, we show that under some conditions on rational functions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918084446080-0836:S0305004124000173:S0305004124000173_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$f_1, ldots, f_nin K(X)$</span></span></img></span></span>, there are only finitely many elements <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918084446080-0836:S0305004124000173:S0305004124000173_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$alpha in K$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918084446080-0836:S0305004124000173:S0305004124000173_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f_1(alpha),ldots,f_n(alpha)$</span></span></img></span></span> are multiplicatively dependent modulo such sets.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves","authors":"ROSS PATERSON","doi":"10.1017/s0305004124000197","DOIUrl":"https://doi.org/10.1017/s0305004124000197","url":null,"abstract":"&lt;p&gt;We show that if &lt;span&gt;F&lt;/span&gt; is &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathbb{Q}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; or a multiquadratic number field, &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$pinleft{{2,3,5}right}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K/F$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is a Galois extension of degree a power of &lt;span&gt;p&lt;/span&gt;, then for elliptic curves &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$E/mathbb{Q}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; ordered by height, the average dimension of the &lt;span&gt;p&lt;/span&gt;-Selmer groups of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$E/K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is bounded. In particular, this provides a bound for the average &lt;span&gt;K&lt;/span&gt;-rank of elliptic curves &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$E/mathbb{Q}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; for such &lt;span&gt;K&lt;/span&gt;. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such &lt;span&gt;F&lt;/span&gt;.&lt;/p&gt;&lt;p&gt;The central result is that: for each finite Galois extension &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K/F$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of number fields and prime number &lt;span&gt;p&lt;/span&gt;, as &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline8.png\"/&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$E/mathbb{Q}$&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; varies, the difference in dimension betwe","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tropical curves in abelian surfaces I: enumeration of curves passing through points","authors":"THOMAS BLOMME","doi":"10.1017/s0305004124000161","DOIUrl":"https://doi.org/10.1017/s0305004124000161","url":null,"abstract":"<p>This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus <span>g</span> curves of fixed degree passing through <span>g</span> points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Differential forms on universal K3 surfaces","authors":"SHOUHEI MA","doi":"10.1017/s0305004124000100","DOIUrl":"https://doi.org/10.1017/s0305004124000100","url":null,"abstract":"We give a vanishing and classification result for holomorphic differential forms on smooth projective models of the moduli spaces of pointed <jats:italic>K</jats:italic>3 surfaces. We prove that there is no nonzero holomorphic <jats:italic>k</jats:italic>-form for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000100_inline1.png\"/> <jats:tex-math> $0&lt;k&lt;10$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and for even <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000100_inline2.png\"/> <jats:tex-math> $k&gt;19$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In the remaining cases, we give an isomorphism between the space of holomorphic <jats:italic>k</jats:italic>-forms with that of vector-valued modular forms (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000100_inline3.png\"/> <jats:tex-math> $10leq k leq 18$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) or scalar-valued cusp forms (odd <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000100_inline4.png\"/> <jats:tex-math> $kgeq 19$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) for the modular group. These results are in fact proved in the generality of lattice-polarisation.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A better than exponent for iterated sums and products over","authors":"OLIVER ROCHE–NEWTON","doi":"10.1017/s0305004124000112","DOIUrl":"https://doi.org/10.1017/s0305004124000112","url":null,"abstract":"In this paper, we prove that the bound &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0305004124000112_eqnU1.png\"/&gt; &lt;jats:tex-math&gt; begin{equation*}max { |8A-7A|,|5f(A)-4f(A)| } gg |A|^{frac{3}{2} + frac{1}{54}}end{equation*} &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt;holds for all &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000112_inline6.png\"/&gt; &lt;jats:tex-math&gt; $A subset mathbb R$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and for all convex functions &lt;jats:italic&gt;f&lt;/jats:italic&gt; which satisfy an additional technical condition. This technical condition is satisfied by the logarithmic function, and this fact can be used to deduce a sum-product estimate &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0305004124000112_eqnU2.png\"/&gt; &lt;jats:tex-math&gt; begin{equation*}max { |16A|, |A^{(16)}| } gg |A|^{frac{3}{2} + c},end{equation*} &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt;for some &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000112_inline7.png\"/&gt; &lt;jats:tex-math&gt; $cgt 0$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Previously, no sum-product estimate over &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000112_inline8.png\"/&gt; &lt;jats:tex-math&gt; $mathbb R$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; with exponent strictly greater than &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000112_inline9.png\"/&gt; &lt;jats:tex-math&gt; $3/2$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; was known for any number of variables. Moreover, the technical condition on &lt;jats:italic&gt;f&lt;/jats:italic&gt; seems to be satisfied for most interesting cases, and we give some further applications. In particular, we show that &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0305004124000112_eqnU3.png\"/&gt; &lt;jats:tex-math&gt; begin{equation*}|AA| leq K|A| implies ,forall d in mathbb R setminus {0 }, ,, |{(a,b) in A times A : a-b=d }| ll K^C |A|^{frac{2}{3}-c^{prime}},end{equation*} &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt;where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000112_inline10.png\"/&gt; &lt;jats:tex-math&gt; $c,C gt 0$ &lt;/jats:tex-math&gt; &lt;/jats:alternat","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Ulam–Hammersley problem for multiset permutations","authors":"LUCAS GERIN","doi":"10.1017/s0305004124000124","DOIUrl":"https://doi.org/10.1017/s0305004124000124","url":null,"abstract":"We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000124_inline1.png\"/> <jats:tex-math> ${1,dots,n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> occurs <jats:italic>k</jats:italic> times, where <jats:italic>k</jats:italic> may depend on <jats:italic>n</jats:italic>. This generalises the famous Ulam–Hammersley problem of the case <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000124_inline2.png\"/> <jats:tex-math> $k=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The proof relies on poissonisation and on a careful non-asymptotic analysis of variants of the Hammersley–Aldous–Diaconis particle system.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"186 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-invertible planar self-affine sets","authors":"ANTTI KÄENMÄKI, PETTERI NISSINEN","doi":"10.1017/s0305004124000136","DOIUrl":"https://doi.org/10.1017/s0305004124000136","url":null,"abstract":"We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"109 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zariski dense surface subgroups in","authors":"CARMEN GALAZ GARCÍA","doi":"10.1017/s0305004124000070","DOIUrl":"https://doi.org/10.1017/s0305004124000070","url":null,"abstract":"<p>For odd <span>n</span> we construct a path <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240325160819198-0562:S0305004124000070:S0305004124000070_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$rho;:;thinspace Pi_1(S) to SL(nmathbb{R})$</span></span></img></span></span> of discrete, faithful, and Zariski dense representations of a surface group such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240325160819198-0562:S0305004124000070:S0305004124000070_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$rho_t(Pi_1(S)) subset SL(n,mathbb{Q})$</span></span></img></span></span> for every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240325160819198-0562:S0305004124000070:S0305004124000070_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$tin mathbb{Q}$</span></span></img></span></span>.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}