The graph of irreducible parabolic subgroups is a combinatorial object associated to an Artin–Tits group A defined so as to coincide with the curve graph of the (n+1) ‐times punctured disk when A is Artin's braid group on (n+1) strands. In this case, it is a hyperbolic graph, by the celebrated Masur–Minsky's theorem. Hyperbolicity of the graph of irreducible parabolic subgroups for more general Artin–Tits groups is an important open question. In this paper, we give a partial affirmative answer.
{"title":"Curve graphs for Artin–Tits groups of type B , A∼ and C∼ are hyperbolic","authors":"M. Calvez, B. A. Cisneros de la Cruz","doi":"10.1112/tlm3.12029","DOIUrl":"https://doi.org/10.1112/tlm3.12029","url":null,"abstract":"The graph of irreducible parabolic subgroups is a combinatorial object associated to an Artin–Tits group A defined so as to coincide with the curve graph of the (n+1) ‐times punctured disk when A is Artin's braid group on (n+1) strands. In this case, it is a hyperbolic graph, by the celebrated Masur–Minsky's theorem. Hyperbolicity of the graph of irreducible parabolic subgroups for more general Artin–Tits groups is an important open question. In this paper, we give a partial affirmative answer.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42902866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the first moment of primes in progressions ∑q⩽x/N(q,a)=1ψ(x;q,a)−xφ(q)as x,N→∞ . We show unconditionally that, when a=1 , there is a significant bias towards negative values, uniformly for N⩽eclogx . The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for a∈Z∖{0} we prove estimates that take into account the potential existence (or inexistence) of Landau–Siegel zeros.
{"title":"The first moment of primes in arithmetic progressions: beyond the Siegel–Walfisz range","authors":"S. Drappeau, D. Fiorilli","doi":"10.1112/tlm3.12030","DOIUrl":"https://doi.org/10.1112/tlm3.12030","url":null,"abstract":"We investigate the first moment of primes in progressions ∑q⩽x/N(q,a)=1ψ(x;q,a)−xφ(q)as x,N→∞ . We show unconditionally that, when a=1 , there is a significant bias towards negative values, uniformly for N⩽eclogx . The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for a∈Z∖{0} we prove estimates that take into account the potential existence (or inexistence) of Landau–Siegel zeros.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41461435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalise to profinite groups some of our previous results on the cohomology of pro‐ p groups of bounded sectional p ‐rank.
我们将先前关于有界截面p秩的亲p群的上同调的一些结果推广到无限群。
{"title":"Cohomology of profinite groups of bounded rank","authors":"P. Symonds","doi":"10.1112/tlm3.12037","DOIUrl":"https://doi.org/10.1112/tlm3.12037","url":null,"abstract":"We generalise to profinite groups some of our previous results on the cohomology of pro‐ p groups of bounded sectional p ‐rank.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47677867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra g of infinitesimal automorphisms of the initial hyper‐Kähler data, we associate a central extension of g , acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one‐loop deformation of the c ‐map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of the corresponding one‐loop deformed c ‐map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous, then the one‐loop deformed metric is of cohomogeneity at most one. As an example, we consider the one‐loop deformation of the symmetric quaternionic Kähler metric on SU(n,2)/S(U(n)×U(2)) , which we prove is of cohomogeneity exactly one. This family generalizes the so‐called universal hypermultiplet ( n=1 ), for which we determine the full isometry group.
{"title":"Symmetries of quaternionic Kähler manifolds with S1 ‐symmetry","authors":"V. Cort'es, A. Saha, D. Thung","doi":"10.1112/tlm3.12026","DOIUrl":"https://doi.org/10.1112/tlm3.12026","url":null,"abstract":"We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra g of infinitesimal automorphisms of the initial hyper‐Kähler data, we associate a central extension of g , acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one‐loop deformation of the c ‐map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of the corresponding one‐loop deformed c ‐map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous, then the one‐loop deformed metric is of cohomogeneity at most one. As an example, we consider the one‐loop deformation of the symmetric quaternionic Kähler metric on SU(n,2)/S(U(n)×U(2)) , which we prove is of cohomogeneity exactly one. This family generalizes the so‐called universal hypermultiplet ( n=1 ), for which we determine the full isometry group.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47333269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let E be a Q ‐isogeny class of elliptic curves defined over Q . The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the Q ‐isogeny class E , and an edge for each cyclic Q ‐isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny‐torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. Then, the main result of the paper is a classification of all the possible isogeny‐torsion graphs that occur for Q ‐isogeny classes of elliptic curves defined over the rationals.
{"title":"A classification of isogeny‐torsion graphs of Q ‐isogeny classes of elliptic curves","authors":"Garen Chiloyan, Álvaro Lozano-Robledo","doi":"10.1112/tlm3.12024","DOIUrl":"https://doi.org/10.1112/tlm3.12024","url":null,"abstract":"Let E be a Q ‐isogeny class of elliptic curves defined over Q . The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in the Q ‐isogeny class E , and an edge for each cyclic Q ‐isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny‐torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. Then, the main result of the paper is a classification of all the possible isogeny‐torsion graphs that occur for Q ‐isogeny classes of elliptic curves defined over the rationals.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46614100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulae which extend the results of Föllmer [Séminaire de probabilités 15 (1981), 143–150] and Cont and Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] to a larger class of functionals, including Föllmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends the Föllmer–Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1‐forms.
我们在停止路径空间上构造了一个新的拓扑，并在该空间的一般域上引入了因果泛函的微积分。我们提出了一种不假设路径变化指数的路径积分的一般方法，并得到了变量的函数变化公式，该公式扩展了Föllmer [ssamminaire de probabilitsamrs 15(1981)， 143-150]和Cont and fourni [J]的结果。功能。《肛门》，259 (2010)，no。[4,1043 - 1072]到更大的泛函类，包括Föllmer的路径积分。我们证明了一类光滑泛函具有鞅性质的路径类似。对于具有有限二次变化的路径，我们的方法扩展了Föllmer-Ito演算，并消除了之前对时间划分序列的限制。我们在这个路径空间上引入了一个叶状结构，并证明调和泛函可以表示为闭合1‐型的路径积分。
{"title":"Causal functional calculus","authors":"H. Chiu, R. Cont","doi":"10.1112/tlm3.12050","DOIUrl":"https://doi.org/10.1112/tlm3.12050","url":null,"abstract":"We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulae which extend the results of Föllmer [Séminaire de probabilités 15 (1981), 143–150] and Cont and Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] to a larger class of functionals, including Föllmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends the Föllmer–Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1‐forms.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Issue Information","authors":"","doi":"10.1112/tlm3.12014","DOIUrl":"https://doi.org/10.1112/tlm3.12014","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46108296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Here we show existence of many subsets of Euclidean spaces that, despite having empty interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure. Most importantly, despite the explicit constructions in our proofs, our methods do not depend on any rectilinear or self‐similar structure of the underlying space. We instead employ the uniform domain condition of Martio and Sarvas. This condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components.
{"title":"Almost uniform domains and Poincaré inequalities","authors":"S. Eriksson-Bique, Jasun Gong","doi":"10.1112/tlm3.12032","DOIUrl":"https://doi.org/10.1112/tlm3.12032","url":null,"abstract":"Here we show existence of many subsets of Euclidean spaces that, despite having empty interior, still support Poincaré inequalities with respect to the restricted Lebesgue measure. Most importantly, despite the explicit constructions in our proofs, our methods do not depend on any rectilinear or self‐similar structure of the underlying space. We instead employ the uniform domain condition of Martio and Sarvas. This condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48331036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fourier‐transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier–Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.
{"title":"Pure point measures with sparse support and sparse Fourier–Bohr support","authors":"M. Baake, Nicolae Strungaru, Venta Terauds","doi":"10.1112/tlm3.12020","DOIUrl":"https://doi.org/10.1112/tlm3.12020","url":null,"abstract":"Fourier‐transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier–Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49360164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let C be a hyperelliptic curve y2=f(x) over a discretely valued field K . The p ‐adic distances between the roots of f(x) can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of C , along with the leading coefficient of f and the action of Gal(K¯/K) on the roots of f , completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of C . In particular, we give an explicit description of the special fibre in terms of this data.
{"title":"Models of hyperelliptic curves with tame potentially semistable reduction","authors":"Omri Faraggi, S. Nowell","doi":"10.1112/tlm3.12023","DOIUrl":"https://doi.org/10.1112/tlm3.12023","url":null,"abstract":"Let C be a hyperelliptic curve y2=f(x) over a discretely valued field K . The p ‐adic distances between the roots of f(x) can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of C , along with the leading coefficient of f and the action of Gal(K¯/K) on the roots of f , completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of C . In particular, we give an explicit description of the special fibre in terms of this data.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45442540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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