{"title":"Riemann曲面的同调群自同构","authors":"R. Hidalgo","doi":"10.17323/1609-4514-2023-23-1-113-120","DOIUrl":null,"url":null,"abstract":"If $\\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\\Gamma'$ is co-compact and torsion free, then $S={\\mathbb H}^{2}/\\Gamma'$ is a closed Riemann surface of genus $g \\geq 2$ admitting the abelian group $A=\\Gamma/\\Gamma'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $\\Gamma_{1}$ and $\\Gamma_{2}$ with $\\Gamma_{1}'=\\Gamma'_{2}$? It is known that if $\\Gamma_{1}$ and $\\Gamma_{2}$ are both of the same signature $(0;k,\\ldots,k)$, for some $k \\geq 2$, then the equality $\\Gamma_{1}'=\\Gamma_{2}'$ ensures that $\\Gamma_{1}=\\Gamma_{2}$. Generalizing this, we observe that if $\\Gamma_{j}$ has signature $(0;k_{j},\\ldots,k_{j})$ and $\\Gamma_{1}'=\\Gamma'_{2}$, then $\\Gamma_{1}=\\Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${\\rm Aut}(S)$ of each homology group $A$ is also obtained.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Homology Group Automorphisms of Riemann Surfaces\",\"authors\":\"R. Hidalgo\",\"doi\":\"10.17323/1609-4514-2023-23-1-113-120\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"If $\\\\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\\\\Gamma'$ is co-compact and torsion free, then $S={\\\\mathbb H}^{2}/\\\\Gamma'$ is a closed Riemann surface of genus $g \\\\geq 2$ admitting the abelian group $A=\\\\Gamma/\\\\Gamma'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $\\\\Gamma_{1}$ and $\\\\Gamma_{2}$ with $\\\\Gamma_{1}'=\\\\Gamma'_{2}$? It is known that if $\\\\Gamma_{1}$ and $\\\\Gamma_{2}$ are both of the same signature $(0;k,\\\\ldots,k)$, for some $k \\\\geq 2$, then the equality $\\\\Gamma_{1}'=\\\\Gamma_{2}'$ ensures that $\\\\Gamma_{1}=\\\\Gamma_{2}$. Generalizing this, we observe that if $\\\\Gamma_{j}$ has signature $(0;k_{j},\\\\ldots,k_{j})$ and $\\\\Gamma_{1}'=\\\\Gamma'_{2}$, then $\\\\Gamma_{1}=\\\\Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${\\\\rm Aut}(S)$ of each homology group $A$ is also obtained.\",\"PeriodicalId\":54736,\"journal\":{\"name\":\"Moscow Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2020-07-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Moscow Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.17323/1609-4514-2023-23-1-113-120\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.17323/1609-4514-2023-23-1-113-120","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
If $\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group $A=\Gamma/\Gamma'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $\Gamma_{1}$ and $\Gamma_{2}$ with $\Gamma_{1}'=\Gamma'_{2}$? It is known that if $\Gamma_{1}$ and $\Gamma_{2}$ are both of the same signature $(0;k,\ldots,k)$, for some $k \geq 2$, then the equality $\Gamma_{1}'=\Gamma_{2}'$ ensures that $\Gamma_{1}=\Gamma_{2}$. Generalizing this, we observe that if $\Gamma_{j}$ has signature $(0;k_{j},\ldots,k_{j})$ and $\Gamma_{1}'=\Gamma'_{2}$, then $\Gamma_{1}=\Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${\rm Aut}(S)$ of each homology group $A$ is also obtained.
期刊介绍:
The Moscow Mathematical Journal (MMJ) is an international quarterly published (paper and electronic) by the Independent University of Moscow and the department of mathematics of the Higher School of Economics, and distributed by the American Mathematical Society. MMJ presents highest quality research and research-expository papers in mathematics from all over the world. Its purpose is to bring together different branches of our science and to achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular.
An important specific trait of the journal is that it especially encourages research-expository papers, which must contain new important results and include detailed introductions, placing the achievements in the context of other studies and explaining the motivation behind the research. The aim is to make the articles — at least the formulation of the main results and their significance — understandable to a wide mathematical audience rather than to a narrow class of specialists.