We study topological groups GG for which either the universal minimal GG-system M(G)M(G) or the universal irreducible affine GG-system IA(G)I!A(G) is tame. We call such groups “intrinsically tame” and “convexly intrinsically tame”, respectively. These notions, which were introduced in [Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Springer, Cham, 2018, pp. 351–392], are generalized versions of extreme amenability and amenability, respectively. When M(G)M(G), as a G
研究了泛极小G G -系统M(G) M(G)或泛不可约仿射G G -系统I A(G) I!A(G)是驯服的拓扑群G G。我们分别称这样的群体为“本质上温顺”和“固有地温顺”。在[遍历理论和动力系统与算术和组合学的相互作用，Springer, Cham, 2018, pp. 351-392]中介绍的这些概念分别是极端易受性和易受性的广义版本。当M(G) M(G)作为一个G -系统，承认循环序时，我们说G本质上是循环序的。这意味着G在本质上是温顺的。我们证明了给定一个圆序集合X°X＿ circ，任何子群G≤a ut (X°)G leq{mathrm Aut{，}(X＿ }circ)对X°X_ circ的作用是超齐次的，当具有点向收敛的拓扑τ p tau ＿p时，本质上是圆有序的。这个结果是Pestov用线性保序变换关于超齐次作用在线性有序集合上的极端可适应性的一个“循环”模拟。对于这样的群G G，我们还描述了系统M(G) M(G)的动力学，证明了它是极近的(因此M(G) M(G)与普遍的强近G -系统重合)，并推导出群G G必须包含一个非阿贝尔自由群。在X X可数的情况下，相应的圆自同构的波兰群G= Aut (X o) G= {mathrm Aut{}，(}X＿o)允许一个具体的描述。利用Kechris-Pestov-Todorcevic结构，我们证明M (G
{"title":"Circular orders, ultra-homogeneous order structures, and their automorphism groups","authors":"E. Glasner, M. Megrelishvili","doi":"10.1090/conm/772/15486","DOIUrl":"https://doi.org/10.1090/conm/772/15486","url":null,"abstract":"<p>We study topological groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for which either the universal minimal <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-system <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">M(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> or the universal irreducible affine <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-system <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I upper A left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mspace width=\"negativethinmathspace\" />\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">I!A(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is tame. We call such groups “intrinsically tame” and “convexly intrinsically tame”, respectively. These notions, which were introduced in [<italic>Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics</italic>, Springer, Cham, 2018, pp. 351–392], are generalized versions of extreme amenability and amenability, respectively. When <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">M(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, as a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation enc","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133995385","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 publication presents some facts and documents related to the biography of the remarkable mathematician Vladimir Abramovich Rokhlin.
本出版物提出了一些事实和文件有关的传记的杰出数学家弗拉基米尔·阿布拉莫维奇·罗克林。
{"title":"V. A. Rokhlin (23 August 1919–3 December 1984), materials for the biography","authors":"A. Vershik","doi":"10.1090/conm/772/15479","DOIUrl":"https://doi.org/10.1090/conm/772/15479","url":null,"abstract":"This publication presents some facts and documents related to the biography of the remarkable mathematician Vladimir Abramovich Rokhlin.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125259709","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 the discreteness of small deformations of a discrete cocompact subgroup of isometries of a locally compact metric space under some natural restrictions.
在一定的自然条件下，证明了局部紧化度量空间的离散紧化等距子群小变形的离散性。
{"title":"Discreteness of deformations of cocompact discrete subgroups","authors":"G. Margulis, G. Soifer","doi":"10.1090/conm/772/15492","DOIUrl":"https://doi.org/10.1090/conm/772/15492","url":null,"abstract":"We prove the discreteness of small deformations of a discrete cocompact subgroup of isometries of a locally compact metric space under some natural restrictions.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131811863","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 article considers the scientific heritage of V. A. Rokhlin in algebraic topology from the point of view of the modern development of mathematics and shows the influence of his results on the development of algebraic topology up to the present. The second part of the article contains new results with fairly detailed sketches of their proofs. There we introduce the notion of partially framed manifolds, which naturally arise in the study of the characteristic classes of vector bundles over the loop space Ω S U ( 2 ) = Ω S P ( 1 ) Omega SU(2)=Omega SP(1) . We obtain theorems on the divisibility of the signature of such manifolds as a result of calculations of characteristic classes with values in complex and quaternionic cobordism.
{"title":"Vladimir Abramovich Rokhlin and algebraic topology","authors":"V. Buchstaber","doi":"10.1090/conm/772/15481","DOIUrl":"https://doi.org/10.1090/conm/772/15481","url":null,"abstract":"The article considers the scientific heritage of V. A. Rokhlin in algebraic topology from the point of view of the modern development of mathematics and shows the influence of his results on the development of algebraic topology up to the present. The second part of the article contains new results with fairly detailed sketches of their proofs. There we introduce the notion of partially framed manifolds, which naturally arise in the study of the characteristic classes of vector bundles over the loop space \u0000\u0000 \u0000 \u0000 Ω\u0000 S\u0000 U\u0000 (\u0000 2\u0000 )\u0000 =\u0000 Ω\u0000 S\u0000 P\u0000 (\u0000 1\u0000 )\u0000 \u0000 Omega SU(2)=Omega SP(1)\u0000 \u0000\u0000. We obtain theorems on the divisibility of the signature of such manifolds as a result of calculations of characteristic classes with values in complex and quaternionic cobordism.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116672734","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":"Rokhlin’s theorem, a problem and a conjecture","authors":"D. Sullivan","doi":"10.1090/conm/772/15497","DOIUrl":"https://doi.org/10.1090/conm/772/15497","url":null,"abstract":"","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124368069","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 talk about several directions of V. Rokhlin’s heritage in ergodic theory: ideas that influenced the further development of investigations (genericity, approximations), problems put forward by V. Rokhlin in his papers, problems that V. Rokhlin put forward verbally (in particular, the question about homogeneous spectrum of finite multiplicity). We touch upon the directions close to the authors of this text and their school. Many of the questions raised by Rokhlin have analogs for different classes of transformations, for group actions, and versions about the genericity of properties appearing in these formulations. We will consider the corresponding topics in such a generalized sense.
{"title":"Group actions: Entropy, mixing, spectra, and generic properties","authors":"A. Stepin, S. Tikhonov","doi":"10.1090/conm/772/15496","DOIUrl":"https://doi.org/10.1090/conm/772/15496","url":null,"abstract":"We talk about several directions of V. Rokhlin’s heritage in ergodic theory: ideas that influenced the further development of investigations (genericity, approximations), problems put forward by V. Rokhlin in his papers, problems that V. Rokhlin put forward verbally (in particular, the question about homogeneous spectrum of finite multiplicity). We touch upon the directions close to the authors of this text and their school. Many of the questions raised by Rokhlin have analogs for different classes of transformations, for group actions, and versions about the genericity of properties appearing in these formulations. We will consider the corresponding topics in such a generalized sense.","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124991890","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}
There are two approaches to the study of the cohomology of group algebras R[G]R[G], the Eilenberg–MacLane cohomology and the Hochschild cohomology. The Eilenberg–MacLane cohomology gives the classical cohomology of the classifying space BGBG (or the Eilenberg–MacLane complex K(G,1)K(G,1)). Note that the space BGBG can be interpreted as a classifying space of the groupoid of the trivial action of the group GG.
The Hochschild cohomology is a more general construction, which considers the so-called bimodules of the algebra R[G]R[G] and their derivative functors
群代数R[G] R[G]的上同调有两种研究方法，即Eilenberg-MacLane上同调和Hochschild上同调。Eilenberg-MacLane上同调给出了分类空间B - G - BG(或Eilenberg-MacLane复合体K(G,1) K(G,1))的经典上同调。注意，空间BG G BG可以被解释为群G的平凡作用群的类群的一个分类空间。Hochschild上同调是一个更一般的构造，它考虑了代数R[G] R[G]及其导数函子Ext(R[G]，M) operatorname Ext{(R[G]，M)的所谓双模，到目前为止还没有任何几何解释。计算Hochschild上同调HH∗(R[G]) HH^*(R[G])的关键是与群G G的伴随作用相关的新类群G R Gr。对于这个类群，相应分类空间BGr BGr的经典上同构与代数r [G] r [G]的Hochschild上同构:H H∗(r [G])≈H f∗(BGr)。}begin{equation*} HH^*(R[G])approx H^*_f(BGr). end{equation*}这一结果对理解群代数的上同调性质的几何性质，特别是对群代数的同调和上同调之间的区别作出了根本性的贡献。本文研究了群代数R[G] R[G]的Hochschild (co)同调群的动机，以及在同调群的支持下，在适当的有限性假设下，用原群G的伴作用群的类群的分类空间的经典(co)同调来描述它。
{"title":"Geometric description of the Hochschild cohomology of group algebras","authors":"A. Mishchenko","doi":"10.1090/conm/772/15494","DOIUrl":"https://doi.org/10.1090/conm/772/15494","url":null,"abstract":"<p>There are two approaches to the study of the cohomology of group algebras <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R left-bracket upper G right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">R[G]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the Eilenberg–MacLane cohomology and the Hochschild cohomology. The Eilenberg–MacLane cohomology gives the classical cohomology of the classifying space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">BG</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (or the Eilenberg–MacLane complex <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K left-parenthesis upper G comma 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K(G,1)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>). Note that the space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">BG</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> can be interpreted as a classifying space of the groupoid of the trivial action of the group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>\u0000\u0000<p>The Hochschild cohomology is a more general construction, which considers the so-called bimodules of the algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R left-bracket upper G right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">R[G]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and their derivative functors <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmln","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125712216","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":"Teaching mathematics to non-mathematicians","authors":"V. Rokhlin","doi":"10.1090/conm/772/15480","DOIUrl":"https://doi.org/10.1090/conm/772/15480","url":null,"abstract":"","PeriodicalId":296603,"journal":{"name":"Topology, Geometry, and Dynamics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130729278","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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