The correspondence principle in physics between quantum mechanics and classical mechanics suggests deep relations between spectral and geometric entities of Riemannian manifolds. We survey---in a way intended to be accessible to a wide audience of mathematicians---a mathematically rigorous instance of such a relation that emerged in recent years, showing a dynamical interpretation of certain Laplace eigenfunctions of hyperbolic surfaces.
{"title":"Dynamics of geodesics, and Maass cusp forms","authors":"A. Pohl, D. Zagier","doi":"10.4171/LEM/66-3/4-2","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-2","url":null,"abstract":"The correspondence principle in physics between quantum mechanics and classical mechanics suggests deep relations between spectral and geometric entities of Riemannian manifolds. We survey---in a way intended to be accessible to a wide audience of mathematicians---a mathematically rigorous instance of such a relation that emerged in recent years, showing a dynamical interpretation of certain Laplace eigenfunctions of hyperbolic surfaces.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116854714","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":"Computable permutations and word problems","authors":"A. Morozov, P. Schupp","doi":"10.4171/LEM/64-1/2-6","DOIUrl":"https://doi.org/10.4171/LEM/64-1/2-6","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130082295","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 prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of saddle connection holonomies, for some Veech surfaces. The proof uses Eisenstein series and relies on earlier work of many authors (notably Selberg). Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.
{"title":"Effective counting for discrete lattice orbits in the plane via Eisenstein series","authors":"Claire Burrin, A. Nevo, Ren'e Ruhr, B. Weiss","doi":"10.4171/LEM/66-3/4-1","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-1","url":null,"abstract":"We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of saddle connection holonomies, for some Veech surfaces. The proof uses Eisenstein series and relies on earlier work of many authors (notably Selberg). Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126316498","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}
L. Bartholdi, Henrika Harer, Thomas Schick Mathematisches Institut, Universitat Gottingen, 'Ecole Normale Sup'erieure, Lyon
We compute the $p$-central and exponent-$p$ series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.
{"title":"Right Angled Artin Groups and partial commutation, old and new","authors":"L. Bartholdi, Henrika Harer, Thomas Schick Mathematisches Institut, Universitat Gottingen, 'Ecole Normale Sup'erieure, Lyon","doi":"10.4171/lem/66-1/2-3","DOIUrl":"https://doi.org/10.4171/lem/66-1/2-3","url":null,"abstract":"We compute the $p$-central and exponent-$p$ series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121286963","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 rationality problems for smooth complete intersections of two quadrics. We focus on the three-dimensional case, with a view toward understanding the invariants governing the rationality of a geometrically rational threefold over a non-closed field.
{"title":"Rationality of complete intersections of two quadrics over nonclosed fields","authors":"B. Hassett, Y. Tschinkel","doi":"10.4171/lem/1001","DOIUrl":"https://doi.org/10.4171/lem/1001","url":null,"abstract":"We study rationality problems for smooth complete intersections of two quadrics. We focus on the three-dimensional case, with a view toward understanding the invariants governing the rationality of a geometrically rational threefold over a non-closed field.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124389399","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 an action selector on aspherical symplectic manifolds that are closed or convex. Such selectors have been constructed by Matthias Schwarz using Floer homology. The construction we present here is simpler and uses only Gromov compactness.
{"title":"A simple construction of an action selector on aspherical symplectic manifolds","authors":"Alberto Abbondandolo, C. Haug, F. Schlenk","doi":"10.4171/LEM/65-1/2-7","DOIUrl":"https://doi.org/10.4171/LEM/65-1/2-7","url":null,"abstract":"We construct an action selector on aspherical symplectic manifolds that are closed or convex. Such selectors have been constructed by Matthias Schwarz using Floer homology. The construction we present here is simpler and uses only Gromov compactness.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"143 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128767394","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 celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3-manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1-vertex triangulations of closed 3-manifolds was independently proven by Matveev and Piergallini. The general result for closed 3-manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3-manifolds. These results (and their proofs) are phrased in the dual language of spines. The purpose of this note is threefold. We wish to popularise Amendola's result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions.
{"title":"Traversing three-manifold triangulations and spines","authors":"J. Rubinstein, Henry Segerman, Stephan Tillmann","doi":"10.4171/lem/65-1/2-5","DOIUrl":"https://doi.org/10.4171/lem/65-1/2-5","url":null,"abstract":"A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3-manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1-vertex triangulations of closed 3-manifolds was independently proven by Matveev and Piergallini. The general result for closed 3-manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3-manifolds. These results (and their proofs) are phrased in the dual language of spines. \u0000The purpose of this note is threefold. We wish to popularise Amendola's result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129383223","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 a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup $G_{bound}$. We give sufficient criteria for triviality and finiteness of $G_{bound}$. We prove that if $G$ is virtually abelian then $G_{bound}$ is finite. In contrast with numerous examples where $G_{bound}$ is trivial, we show that for every finite group $A$, there exists an infinite group $G$ with $G_{bound}=A$. This group $G$ can be chosen among torsion groups. We also study the group $G_{bound}(d)$ of elements with uniformly bounded word-length for generating sets of cardinality less than $d$.
{"title":"Elements of uniformly bounded word-length in groups","authors":"Yanis Amirou","doi":"10.4171/lem/1002","DOIUrl":"https://doi.org/10.4171/lem/1002","url":null,"abstract":"We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup $G_{bound}$. We give sufficient criteria for triviality and finiteness of $G_{bound}$. We prove that if $G$ is virtually abelian then $G_{bound}$ is finite. In contrast with numerous examples where $G_{bound}$ is trivial, we show that for every finite group $A$, there exists an infinite group $G$ with $G_{bound}=A$. This group $G$ can be chosen among torsion groups. We also study the group $G_{bound}(d)$ of elements with uniformly bounded word-length for generating sets of cardinality less than $d$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132133252","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}
The purpose of this note is to prove the $G$-equivariant Sarkisov program for a connected algebraic group $G$ following the proof of the Sarkisov program by Hacon and McKernan. As a consequence, we obtain a characterisation of connected subgroups of $Bir(Z)$ acting rationally on $Z$.
{"title":"A note on the $G$-Sarkisov program","authors":"E. Floris","doi":"10.4171/lem/66-1/2-5","DOIUrl":"https://doi.org/10.4171/lem/66-1/2-5","url":null,"abstract":"The purpose of this note is to prove the $G$-equivariant Sarkisov program for a connected algebraic group $G$ following the proof of the Sarkisov program by Hacon and McKernan. As a consequence, we obtain a characterisation of connected subgroups of $Bir(Z)$ acting rationally on $Z$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115026511","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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