Pierre de la Harpe, Anders Karlsson, T. Nagnibeda, F. Rădulescu
{"title":"Vaughan Jones (1952–2020)","authors":"Pierre de la Harpe, Anders Karlsson, T. Nagnibeda, F. Rădulescu","doi":"10.4171/lem/1044","DOIUrl":"https://doi.org/10.4171/lem/1044","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"143 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122870097","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":"Commission Internationale de l’Enseignement Mathématique. Once upon a time . . . Historical vignettes from the Archives of ICMI: The Aarhus ICMI-seminar on the teaching of geometry","authors":"Bernard R. Hodgson","doi":"10.4171/lem/1039","DOIUrl":"https://doi.org/10.4171/lem/1039","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134599958","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":"Commission Internationale de l’Enseignement Mathématique. Once upon a time ... Historical vignettes from the Archives of ICMI: The IMU centennial","authors":"Bernard R. Hodgson","doi":"10.4171/lem/1028","DOIUrl":"https://doi.org/10.4171/lem/1028","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"133 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130998539","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 give an account of the Planarity Theorem of Maskit. This gives a classification of finitely generated groups acting effectively properly discontinuously by orientation-preserving homeomorphisms on a planar surface. One can also realise such groups as kleinian function groups. We also explain how one can give another proof of the planarity theorem using Dunwoody’s theory of tracks.
{"title":"Notes on Maskit’s Planarity Theorem","authors":"B. Bowditch","doi":"10.4171/lem/1019","DOIUrl":"https://doi.org/10.4171/lem/1019","url":null,"abstract":". We give an account of the Planarity Theorem of Maskit. This gives a classification of finitely generated groups acting effectively properly discontinuously by orientation-preserving homeomorphisms on a planar surface. One can also realise such groups as kleinian function groups. We also explain how one can give another proof of the planarity theorem using Dunwoody’s theory of tracks.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130991052","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}
. Given a finite group of automorphisms of a compact Riemann sur- face, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.
{"title":"Residues of connections and the Chevalley–Weil formula for curves","authors":"D. Arapura","doi":"10.4171/lem/1030","DOIUrl":"https://doi.org/10.4171/lem/1030","url":null,"abstract":". Given a finite group of automorphisms of a compact Riemann sur- face, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128360751","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 define and study a new compactification, called the height compactification of the horospheric product of two infinite trees. We will provide a complete description of this compactification. In particular, we show that this compactification is isomorphic to the Busemann compactification when all the vertices of both trees have degree at least three, which also leads to a precise description of the Busemann functions in terms of the points in the geometric compactification of each tree. We will discuss an application to the asymptotic behavior of integrable ergodic cocycles with values in the isometry group of such horospheric product.
{"title":"Compactifications of horospheric products","authors":"Behrang Forghani, Keivan Mallahi-Karai","doi":"10.4171/LEM/1025","DOIUrl":"https://doi.org/10.4171/LEM/1025","url":null,"abstract":". We define and study a new compactification, called the height compactification of the horospheric product of two infinite trees. We will provide a complete description of this compactification. In particular, we show that this compactification is isomorphic to the Busemann compactification when all the vertices of both trees have degree at least three, which also leads to a precise description of the Busemann functions in terms of the points in the geometric compactification of each tree. We will discuss an application to the asymptotic behavior of integrable ergodic cocycles with values in the isometry group of such horospheric product.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123190974","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 explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {em binary subgroups}, $B(Sigma,mu)
{"title":"Binary subgroups of direct products","authors":"M. Bridson","doi":"10.4171/lem/1057","DOIUrl":"https://doi.org/10.4171/lem/1057","url":null,"abstract":"We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {em binary subgroups}, $B(Sigma,mu)<G_1timesdotstimes G_m$. These full subdirect products require strikingly few generators. If each $G_i$ is finitely presented, $B(Sigma,mu)$ is finitely presented. When the $G_i$ are non-abelian limit groups (e.g. free or surface groups), the $B(Sigma,mu)$ provide new examples of finitely presented, residually-free groups that do not have finite classifying spaces and are not of Stallings-Bieri type. These examples settle a question of Minasyan relating different notions of rank for residually-free groups. Using binary subgroups, we prove that if $G_1,dots,G_m$ are perfect groups, each requiring at most $r$ generators, then $G_1timesdotstimes G_m$ requires at most $r lfloor log_2 m+1 rfloor$ generators.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123938368","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}
For certain generating sets of the subfactor pair $Msubset Mrtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $Mrtimes(mathbb{Z}/2mathbb{Z})^{oplusinfty}$ for a large class of algebras $M$.
{"title":"An inequality for non-microstates free entropy dimension for crossed products by finite abelian groups","authors":"D. Shlyakhtenko","doi":"10.4171/lem/1056","DOIUrl":"https://doi.org/10.4171/lem/1056","url":null,"abstract":"For certain generating sets of the subfactor pair $Msubset Mrtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $Mrtimes(mathbb{Z}/2mathbb{Z})^{oplusinfty}$ for a large class of algebras $M$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134006262","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 bispectral problem is motivated by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admitting a second-order differential operator with a simple spectrum in its commutator. In this article, we discuss a noncommutative version of the bispectral problem, obtained by allowing all objects in the original formulation to be matrix-valued. Deep attention is given to bispectral algebras and their presentations as a tool to get information about bispectral triples.
{"title":"Matrix bispectrality and noncommutative algebras: Beyond the prolate spheroidals","authors":"F. Grunbaum, Brian D. Vasquez, J. Zubelli","doi":"10.4171/lem/1053","DOIUrl":"https://doi.org/10.4171/lem/1053","url":null,"abstract":"The bispectral problem is motivated by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admitting a second-order differential operator with a simple spectrum in its commutator. In this article, we discuss a noncommutative version of the bispectral problem, obtained by allowing all objects in the original formulation to be matrix-valued. Deep attention is given to bispectral algebras and their presentations as a tool to get information about bispectral triples.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133956764","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 that an outer action of a locally compact group $G$ on a full factor $M$ is automatically strictly outer, meaning that the relative commutant of $M$ in the crossed product is trivial. If moreover the image of $G$ in the outer automorphism group $operatorname{Out} M$ is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of $M$ in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical Bogoljubov actions on free group factors or free Araki-Woods factors, these results are new.
{"title":"Spectral gap and strict outerness for actions of locally compact groups on full factors","authors":"A. Marrakchi, S. Vaes","doi":"10.4171/lem/1054","DOIUrl":"https://doi.org/10.4171/lem/1054","url":null,"abstract":"We prove that an outer action of a locally compact group $G$ on a full factor $M$ is automatically strictly outer, meaning that the relative commutant of $M$ in the crossed product is trivial. If moreover the image of $G$ in the outer automorphism group $operatorname{Out} M$ is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of $M$ in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical Bogoljubov actions on free group factors or free Araki-Woods factors, these results are new.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126230009","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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