We consider the moduli space N of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle A → N of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections — local holomorphic sections of A — that allow to pull back the Liouville symplectic form on T N to A . We prove that the pullback of the Goldman form to A by the Riemann-Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to N by the Narasimhan-Seshadri connection is the natural symplectic form on N , introduced by Narasimhan and Atiyah & Bott.
{"title":"Goldman form, flat connections and stable vector bundles","authors":"L. Takhtajan","doi":"10.4171/lem/1036","DOIUrl":"https://doi.org/10.4171/lem/1036","url":null,"abstract":"We consider the moduli space N of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle A → N of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections — local holomorphic sections of A — that allow to pull back the Liouville symplectic form on T N to A . We prove that the pullback of the Goldman form to A by the Riemann-Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to N by the Narasimhan-Seshadri connection is the natural symplectic form on N , introduced by Narasimhan and Atiyah & Bott.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115055059","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 some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S={2^{m}3^{n}}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $Lambda(p)$ set for all $2
{"title":"The Furstenberg set and its random version","authors":"A. Fan, Herv'e Queff'elec, M. Queff'elec","doi":"10.4171/lem/1040","DOIUrl":"https://doi.org/10.4171/lem/1040","url":null,"abstract":"We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S={2^{m}3^{n}}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is \"Khinchin distributed\", is far from being Hartman-distributed while $T$ is, and that $S$ is a $Lambda(p)$ set for all $2<p<infty$ and that $T$ is a $p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116181156","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 provide obstructions to a link in S3 arising as the cross section of any number of unlinked spheres in S4. Our obstructions arise from the multivariable signature, the Blanchfield form and generalised Seifert matrices. We also obtain obstructions in the case of surfaces of higher genera, leading to a lower bound on the doubly slice genus of links.
{"title":"Abelian invariants of doubly slice links","authors":"Anthony Conway, P. Orson","doi":"10.4171/lem/1029","DOIUrl":"https://doi.org/10.4171/lem/1029","url":null,"abstract":"We provide obstructions to a link in S3 arising as the cross section of any number of unlinked spheres in S4. Our obstructions arise from the multivariable signature, the Blanchfield form and generalised Seifert matrices. We also obtain obstructions in the case of surfaces of higher genera, leading to a lower bound on the doubly slice genus of links.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115645550","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}
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.
{"title":"The commutator subgroups of free groups and surface groups","authors":"Andrew Putman","doi":"10.4171/lem/1035","DOIUrl":"https://doi.org/10.4171/lem/1035","url":null,"abstract":"A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129145554","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}
This is a largely expository paper, providing a self-contained account on the results of [Sch-Si1, Sch-Si2], in the cases denoted there 2Q and 2M. These papers of Schäfke and Singer supplied new proofs to the main theorems of [Bez-Bou, Ad-Be], on the rationality of power series satisfying a pair of independent q-difference, or Mahler, equations. We emphasize the language of Γ-difference modules, instead of difference equations or systems. Although in the two cases mentioned above this is only a semantic change, we also treat a new case, which may be labeled 1M1Q. Here the group Γ is generalized dihedral rather than abelian, and the language of equations is inadequate. In the last section we explain how to generalize the main theorems in case 2Q to finite characteristic.
{"title":"On the structure of certain $Gamma$-difference modules","authors":"E. D. Shalit, J. Guti'errez","doi":"10.4171/lem/1032","DOIUrl":"https://doi.org/10.4171/lem/1032","url":null,"abstract":"This is a largely expository paper, providing a self-contained account on the results of [Sch-Si1, Sch-Si2], in the cases denoted there 2Q and 2M. These papers of Schäfke and Singer supplied new proofs to the main theorems of [Bez-Bou, Ad-Be], on the rationality of power series satisfying a pair of independent q-difference, or Mahler, equations. We emphasize the language of Γ-difference modules, instead of difference equations or systems. Although in the two cases mentioned above this is only a semantic change, we also treat a new case, which may be labeled 1M1Q. Here the group Γ is generalized dihedral rather than abelian, and the language of equations is inadequate. In the last section we explain how to generalize the main theorems in case 2Q to finite characteristic.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132065203","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}
Le texte qui suit passe en revue les propriétés des groupes de Coxeter finis qui sont les plus utiles pour la compréhension de leurs invariants cohomologiques, au sens de [Se 03]. La plupart de ces propriétés se trouvent déjà dans la littérature (par exemple [Bo 68], [Ca 72], [Sp 74], [Sp 82], [Hu 90], [Ka 01], [DPR 13]), mais il a paru commode de les rassembler de façon systématique, et de les compléter sur quelques points. Il s’agit surtout des involutions (§2), car ce sont leurs classes de conjugaison qui paramètrent de façon naturelle les invariants cohomologiques du groupe, cf. [Se 18]. Elles interviennent le plus souvent par l’intermédiaire des sous-groupes appelés “cubes” : sous-groupes abéliens engendrés par des réflexions (§4). Ces groupes jouent un rôle analogue à celui des groupes de Sylow, cf. par exemple th.4.16. Leur intérêt pour la détermination des invariants cohomologiques provient du “splitting principle” : sous certaines conditions techniques, un invariant cohomologique est nul si ses restrictions à tous les cubes sont nulles, cf. [Se 03], [Hi 10], [Se 18], [Hi 20], [GH 21]. Le §5 décrit les différents types de groupes irréductibles, en insistant sur les propriétés de leurs involutions et de leurs cubes, notamment pour les types Bn, Dn, E7 et E8. Il contient aussi une liste d’inclusions entre différents types qui se révèle utile pour prouver certains cas du “splitting principle”, comme nous le montrerons ailleurs. Le §6 est consacré à la construction, et aux propriétés, de certains groupes de Coxeter de rang 4, notamment ceux de type F4 et H4, cf. th.6.12.
{"title":"Groupes de Coxeter finis: involutions et cubes","authors":"Jean-Pierre Serre","doi":"10.4171/lem/1022","DOIUrl":"https://doi.org/10.4171/lem/1022","url":null,"abstract":"Le texte qui suit passe en revue les propriétés des groupes de Coxeter finis qui sont les plus utiles pour la compréhension de leurs invariants cohomologiques, au sens de [Se 03]. La plupart de ces propriétés se trouvent déjà dans la littérature (par exemple [Bo 68], [Ca 72], [Sp 74], [Sp 82], [Hu 90], [Ka 01], [DPR 13]), mais il a paru commode de les rassembler de façon systématique, et de les compléter sur quelques points. Il s’agit surtout des involutions (§2), car ce sont leurs classes de conjugaison qui paramètrent de façon naturelle les invariants cohomologiques du groupe, cf. [Se 18]. Elles interviennent le plus souvent par l’intermédiaire des sous-groupes appelés “cubes” : sous-groupes abéliens engendrés par des réflexions (§4). Ces groupes jouent un rôle analogue à celui des groupes de Sylow, cf. par exemple th.4.16. Leur intérêt pour la détermination des invariants cohomologiques provient du “splitting principle” : sous certaines conditions techniques, un invariant cohomologique est nul si ses restrictions à tous les cubes sont nulles, cf. [Se 03], [Hi 10], [Se 18], [Hi 20], [GH 21]. Le §5 décrit les différents types de groupes irréductibles, en insistant sur les propriétés de leurs involutions et de leurs cubes, notamment pour les types Bn, Dn, E7 et E8. Il contient aussi une liste d’inclusions entre différents types qui se révèle utile pour prouver certains cas du “splitting principle”, comme nous le montrerons ailleurs. Le §6 est consacré à la construction, et aux propriétés, de certains groupes de Coxeter de rang 4, notamment ceux de type F4 et H4, cf. th.6.12.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126678165","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 the topological structure and the topological dynamics of groups of homeomorphisms of scattered spaces. For a large class of them (including the homeomorphism group of any ordinal space or of any locally compact scattered space), we establish Roelcke-precompactness and amenability, classify all closed normal subgroups and compute the universal minimal flow. As a by-product, we classify up to isomorphism the homeomorphism groups of compact ordinal spaces.
{"title":"Dynamics and structure of groups of homeomorphisms of scattered spaces","authors":"Maxime Gheysens","doi":"10.4171/lem/1007","DOIUrl":"https://doi.org/10.4171/lem/1007","url":null,"abstract":"We study the topological structure and the topological dynamics of groups of homeomorphisms of scattered spaces. For a large class of them (including the homeomorphism group of any ordinal space or of any locally compact scattered space), we establish Roelcke-precompactness and amenability, classify all closed normal subgroups and compute the universal minimal flow. As a by-product, we classify up to isomorphism the homeomorphism groups of compact ordinal spaces.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126395217","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}
M. Bestvina, Ryan Dickmann, G. Domat, Sanghoon Kwak, Priyam Patel, Emily Stark
The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten their proofs.
{"title":"Free products from spinning and rotating families","authors":"M. Bestvina, Ryan Dickmann, G. Domat, Sanghoon Kwak, Priyam Patel, Emily Stark","doi":"10.4171/lem/1033","DOIUrl":"https://doi.org/10.4171/lem/1033","url":null,"abstract":"The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten their proofs.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124600979","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}
A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space intersecting any convex body of volume one. A suitable weakening of the volume constraint leads to the (much more recent) problem of constructing emph{dense forests}. These are discrete point sets getting uniformly close to long enough line segments. Progress towards these problems have so far involved a wide range of ideas surrounding areas as varied as combinatorial and computation geometry, convex geometry, Diophantine approximation, discrepancy theory, the theory of dynamical systems, the theory of exponential sums, Fourier analysis, homogeneous dynamics, the mathematical theory of quasicrystals and probability theory. The goal of this paper is to survey the known results related to the Danzer Problem and to the construction of dense forests, to generalise some of them and to state a number of open problems to make further progress towards a solution to this longstanding question.
{"title":"Around the Danzer problem and the construction of dense forests","authors":"F. Adiceam","doi":"10.4171/lem/1020","DOIUrl":"https://doi.org/10.4171/lem/1020","url":null,"abstract":"A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space intersecting any convex body of volume one. A suitable weakening of the volume constraint leads to the (much more recent) problem of constructing emph{dense forests}. These are discrete point sets getting uniformly close to long enough line segments. \u0000Progress towards these problems have so far involved a wide range of ideas surrounding areas as varied as combinatorial and computation geometry, convex geometry, Diophantine approximation, discrepancy theory, the theory of dynamical systems, the theory of exponential sums, Fourier analysis, homogeneous dynamics, the mathematical theory of quasicrystals and probability theory. \u0000The goal of this paper is to survey the known results related to the Danzer Problem and to the construction of dense forests, to generalise some of them and to state a number of open problems to make further progress towards a solution to this longstanding question.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115590294","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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