We study several natural decision problems in braid groups and Artin groups. We classify the Artin groups with decidable submonoid membership problem in terms of the non-existence of certain forbidden induced subgraphs of the defining graph. Furthermore, we also classify the Artin groups for which the following problems are decidable: the rational subset membership problem, semigroup intersection problem, fixed-target submonoid membership problem, and the rational identity problem. In the case of braid groups our results show that the submonoid membership problem, and each and every one of these problems, is decidable in the braid group $mathbf{B}_n$ if and only if $n leq 3$, which answers an open problem of Potapov (2013). Our results also generalize and extend results of Lohrey & Steinberg (2008) who classified right-angled Artin groups with decidable submonoid (and rational subset) membership problem.
{"title":"Membership problems in braid groups and Artin groups","authors":"Robert D. Gray, Carl-Fredrik Nyberg-Brodda","doi":"arxiv-2409.11335","DOIUrl":"https://doi.org/arxiv-2409.11335","url":null,"abstract":"We study several natural decision problems in braid groups and Artin groups.\u0000We classify the Artin groups with decidable submonoid membership problem in\u0000terms of the non-existence of certain forbidden induced subgraphs of the\u0000defining graph. Furthermore, we also classify the Artin groups for which the\u0000following problems are decidable: the rational subset membership problem,\u0000semigroup intersection problem, fixed-target submonoid membership problem, and\u0000the rational identity problem. In the case of braid groups our results show\u0000that the submonoid membership problem, and each and every one of these\u0000problems, is decidable in the braid group $mathbf{B}_n$ if and only if $n leq\u00003$, which answers an open problem of Potapov (2013). Our results also\u0000generalize and extend results of Lohrey & Steinberg (2008) who classified\u0000right-angled Artin groups with decidable submonoid (and rational subset)\u0000membership problem.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253747","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 Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite simple non-abelian group $G$ and any set $Asubseteq G$ with $|A|geq 2$, $G$ is the product of at most $Nfrac{log|G|}{log|A|}$ conjugates of $A$, for some absolute constant $N$. For $G$ of Lie type, we prove that for any $varepsilon>0$ there is some $N_{varepsilon}$ for which $G$ is the product of at most $N_{varepsilon}left(frac{log|G|}{log|A|}right)^{1+varepsilon}$ conjugates of either $A$ or $A^{-1}$. For symmetric sets, this improves on results of Liebeck, Nikolov, and Shalev [LNS12] and Gill, Pyber, Short, and Szab'o [GPSS13]. During the preparation of this paper, the proof of the Liebeck-Nikolov-Shalev conjecture was completed by Lifshitz [Lif24]. Both papers use [GLPS24] as a starting point. Lifshitz's argument uses heavy machinery from representation theory to complete the conjecture, whereas this paper achieves a more modest result by rather elementary combinatorial arguments.
{"title":"Writing finite simple groups of Lie type as products of subset conjugates","authors":"Daniele Dona","doi":"arxiv-2409.11246","DOIUrl":"https://doi.org/arxiv-2409.11246","url":null,"abstract":"The Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite\u0000simple non-abelian group $G$ and any set $Asubseteq G$ with $|A|geq 2$, $G$\u0000is the product of at most $Nfrac{log|G|}{log|A|}$ conjugates of $A$, for\u0000some absolute constant $N$. For $G$ of Lie type, we prove that for any $varepsilon>0$ there is some\u0000$N_{varepsilon}$ for which $G$ is the product of at most\u0000$N_{varepsilon}left(frac{log|G|}{log|A|}right)^{1+varepsilon}$\u0000conjugates of either $A$ or $A^{-1}$. For symmetric sets, this improves on\u0000results of Liebeck, Nikolov, and Shalev [LNS12] and Gill, Pyber, Short, and\u0000Szab'o [GPSS13]. During the preparation of this paper, the proof of the Liebeck-Nikolov-Shalev\u0000conjecture was completed by Lifshitz [Lif24]. Both papers use [GLPS24] as a\u0000starting point. Lifshitz's argument uses heavy machinery from representation\u0000theory to complete the conjecture, whereas this paper achieves a more modest\u0000result by rather elementary combinatorial arguments.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253746","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}
María José Felipe, María Dolores Pérez-Ramos, Víctor Sotomayor
Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the $G$-orbits of irreducible characters of $N$. In the present paper, we provide an alternative approach to this fact through the structure of the group algebra. We also show that such matrices are non-singular and become a useful tool to obtain information of $N$ from the character table of $G$.
{"title":"On $G$-character tables for normal subgroups","authors":"María José Felipe, María Dolores Pérez-Ramos, Víctor Sotomayor","doi":"arxiv-2409.11591","DOIUrl":"https://doi.org/arxiv-2409.11591","url":null,"abstract":"Let $N$ be a normal subgroup of a finite group $G$. From a result due to\u0000Brauer, it can be derived that the character table of $G$ contains square\u0000submatrices which are induced by the $G$-conjugacy classes of elements in $N$\u0000and the $G$-orbits of irreducible characters of $N$. In the present paper, we\u0000provide an alternative approach to this fact through the structure of the group\u0000algebra. We also show that such matrices are non-singular and become a useful\u0000tool to obtain information of $N$ from the character table of $G$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253753","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 two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$ and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem 1.1). Furthermore, if $G$ is a prosoluble group in which for every subset $pisubseteqpi(G)$ there is a Hall $pi$-subgroup $H_pi$ and a Hall $pi'$-subgroup $H_{pi'}$ such that $Pr(H_pi,H_{pi'})>0$, then $G$ is virtually pronilpotent (Theorem 1.2).
{"title":"Commuting probability for the Sylow subgroups of a profinite group","authors":"Eloisa Detomi, Marta Morigi, Pavel Shumyatsky","doi":"arxiv-2409.11165","DOIUrl":"https://doi.org/arxiv-2409.11165","url":null,"abstract":"Given two subgroups $H,K$ of a compact group $G$, the probability that a\u0000random element of $H$ commutes with a random element of $K$ is denoted by\u0000$Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$,\u0000a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$\u0000and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem\u00001.1). Furthermore, if $G$ is a prosoluble group in which for every subset\u0000$pisubseteqpi(G)$ there is a Hall $pi$-subgroup $H_pi$ and a Hall\u0000$pi'$-subgroup $H_{pi'}$ such that $Pr(H_pi,H_{pi'})>0$, then $G$ is\u0000virtually pronilpotent (Theorem 1.2).","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253748","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}
In this paper, we establish the growth tightness of the quotient by confined subgroups in groups admitting the statistically convex-cocompact action with contracting elements. The result is sharp in the sense that the actions could not be relaxed with purely exponential growth. Applications to uniformly recurrent subgroups are discussed.
{"title":"Growth tightness of quotients by confined subgroups","authors":"Lihuang Ding, Wenyuan Yang","doi":"arxiv-2409.10268","DOIUrl":"https://doi.org/arxiv-2409.10268","url":null,"abstract":"In this paper, we establish the growth tightness of the quotient by confined\u0000subgroups in groups admitting the statistically convex-cocompact action with\u0000contracting elements. The result is sharp in the sense that the actions could\u0000not be relaxed with purely exponential growth. Applications to uniformly\u0000recurrent subgroups are discussed.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253751","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}
In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $mathbb{Q}$-defined linear algebraic group $mathbf{G}$, there exists an isomorphism between the cohomology rings with coefficients in a finite dimensional rational $mathbf{G}$-module $M$ of the associated $mathbb{Q}$-defined Lie algebra $mathfrak{g_mathbb{Q}}$ and Zariski dense subgroups $Gamma leq mathbf{G}(mathbb{Q})$ that satisfy the condition that they intersect the $mathbb{Q}$-split maximal torus discretely. We further prove that the restriction map in rational cohomology from $mathbf{G}$ to a Zariski dense subgroup $Gamma leq mathbf{G}(mathbb{Q})$ with coefficients in $M$ is an injection. We then derive several results regarding finitely generated solvable groups of finite abelian rank and their representations on cohomology.
{"title":"Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups","authors":"Milana Golich, Mark Pengitore","doi":"arxiv-2409.09987","DOIUrl":"https://doi.org/arxiv-2409.09987","url":null,"abstract":"In this article, we establish results concerning the cohomology of Zariski\u0000dense subgroups of solvable linear algebraic groups. We show that for an\u0000irreducible solvable $mathbb{Q}$-defined linear algebraic group $mathbf{G}$,\u0000there exists an isomorphism between the cohomology rings with coefficients in a\u0000finite dimensional rational $mathbf{G}$-module $M$ of the associated\u0000$mathbb{Q}$-defined Lie algebra $mathfrak{g_mathbb{Q}}$ and Zariski dense\u0000subgroups $Gamma leq mathbf{G}(mathbb{Q})$ that satisfy the condition that\u0000they intersect the $mathbb{Q}$-split maximal torus discretely. We further\u0000prove that the restriction map in rational cohomology from $mathbf{G}$ to a\u0000Zariski dense subgroup $Gamma leq mathbf{G}(mathbb{Q})$ with coefficients\u0000in $M$ is an injection. We then derive several results regarding finitely\u0000generated solvable groups of finite abelian rank and their representations on\u0000cohomology.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253756","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}
Neeraj Kumar Dhanwani, Pravin Kumar, Tushar Kanta Naik, Mahender Singh
Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the well-known virtual braid groups. For each Coxeter graph $Gamma$, they defined the virtual Artin group $VA[Gamma]$, which is generated by the corresponding Artin group $A[Gamma]$ and the Coxeter group $W[Gamma]$, subject to certain mixed relations inspired by the action of $W[Gamma]$ on its root system $Phi[Gamma]$. There is a natural surjection $ mathrm{VA}[Gamma] rightarrow W[Gamma]$, with the kernel $PVA[Gamma]$ representing the pure virtual Artin group. In this paper, we explore linearity, crystallographic quotients, and automorphisms of certain classes of virtual Artin groups. Inspired from the work of Cohen, Wales, and Krammer, we construct a linear representation of the virtual Artin group $VA[Gamma]$. As a consequence of this representation, we deduce that if $W[Gamma]$ is a spherical Coxeter group, then $VA[Gamma]/PVA[Gamma]'$ is a crystallographic group of dimension $ |Phi[Gamma]|$ with the holonomy group $W[Gamma]$. Further, extending an idea of Davis and Januszkiewicz, we prove that all right-angled virtual Artin groups admit a faithful linear representation. The remainder of the paper focuses on conjugacy classes and automorphisms of a subclass of right-angled virtual Artin groups, $VAT_n$, associated with planar braid groups called twin groups. We determine the automorphism group of $VAT_n$ for each $ngeq 5$, and give a precise description of a generic automorphism. As an application of this description, we prove that $VAT_n$ has the $R_infty$-property for each $n ge 2$.
{"title":"Linearity, crystallographic quotients, and automorphisms of virtual Artin groups","authors":"Neeraj Kumar Dhanwani, Pravin Kumar, Tushar Kanta Naik, Mahender Singh","doi":"arxiv-2409.10270","DOIUrl":"https://doi.org/arxiv-2409.10270","url":null,"abstract":"Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel\u0000as broad generalizations of the well-known virtual braid groups. For each\u0000Coxeter graph $Gamma$, they defined the virtual Artin group $VA[Gamma]$,\u0000which is generated by the corresponding Artin group $A[Gamma]$ and the Coxeter\u0000group $W[Gamma]$, subject to certain mixed relations inspired by the action of\u0000$W[Gamma]$ on its root system $Phi[Gamma]$. There is a natural surjection $\u0000mathrm{VA}[Gamma] rightarrow W[Gamma]$, with the kernel $PVA[Gamma]$\u0000representing the pure virtual Artin group. In this paper, we explore linearity,\u0000crystallographic quotients, and automorphisms of certain classes of virtual\u0000Artin groups. Inspired from the work of Cohen, Wales, and Krammer, we construct\u0000a linear representation of the virtual Artin group $VA[Gamma]$. As a\u0000consequence of this representation, we deduce that if $W[Gamma]$ is a\u0000spherical Coxeter group, then $VA[Gamma]/PVA[Gamma]'$ is a crystallographic\u0000group of dimension $ |Phi[Gamma]|$ with the holonomy group $W[Gamma]$.\u0000Further, extending an idea of Davis and Januszkiewicz, we prove that all\u0000right-angled virtual Artin groups admit a faithful linear representation. The\u0000remainder of the paper focuses on conjugacy classes and automorphisms of a\u0000subclass of right-angled virtual Artin groups, $VAT_n$, associated with planar\u0000braid groups called twin groups. We determine the automorphism group of $VAT_n$\u0000for each $ngeq 5$, and give a precise description of a generic automorphism.\u0000As an application of this description, we prove that $VAT_n$ has the\u0000$R_infty$-property for each $n ge 2$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253750","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}
Jesús Alonso Ochoa Arango, María Angélica Umbarila Martín
In this work, we study the function $f_2(G)$ that counts the number of exact factorizations of a finite group $G$. We compute $f_2(G)$ for some well-known families of finite groups and use the results of Wiegold and Williamson cite{WW} to derive an asymptotic expression for the number of exact factorizations of the alternating group $A_{2^n}$. Finally, we propose several questions about the function $f_2(G)$ that may be of interest for further research.
{"title":"On the number of exact factorization of finite Groups","authors":"Jesús Alonso Ochoa Arango, María Angélica Umbarila Martín","doi":"arxiv-2409.10428","DOIUrl":"https://doi.org/arxiv-2409.10428","url":null,"abstract":"In this work, we study the function $f_2(G)$ that counts the number of exact\u0000factorizations of a finite group $G$. We compute $f_2(G)$ for some well-known\u0000families of finite groups and use the results of Wiegold and Williamson\u0000cite{WW} to derive an asymptotic expression for the number of exact\u0000factorizations of the alternating group $A_{2^n}$. Finally, we propose several\u0000questions about the function $f_2(G)$ that may be of interest for further\u0000research.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253749","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 group $G$ is called a Howson group if the intersection $Hcap K$ of any two finitely generated subgroups $H, K
如果任意两个有限生成的子群 $H，K
{"title":"Howson groups which are not strongly Howson","authors":"Qiang Zhang, Dongxiao Zhao","doi":"arxiv-2409.09567","DOIUrl":"https://doi.org/arxiv-2409.09567","url":null,"abstract":"A group $G$ is called a Howson group if the intersection $Hcap K$ of any two\u0000finitely generated subgroups $H, K<G$ is again finitely generated, and called a\u0000strongly Howson group when a uniform bound for the rank of $Hcap K$ can be\u0000obtained from the ranks of $H$ and $K$. Clearly, every strongly Howson group is\u0000a Howson group, but it is unclear in the literature whether the converse is\u0000true. In this note, we show that the converse is not true by constructing the\u0000first Howson groups which are not strongly Howson.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253758","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 a prime $p$, fusion systems over discrete $p$-toral groups are categories that model and generalize the $p$-local structure of Lie groups and certain other infinite groups in the same way that fusion systems over finite $p$-groups model and generalize the $p$-local structure of finite groups. In the finite case, it is natural to say that a fusion system $mathcal{F}$ is realizable if it is isomorphic to the fusion system of a finite group, but it is less clear what realizability should mean in the discrete $p$-toral case. In this paper, we look at some of the different types of realizability for fusion systems over discrete $p$-toral groups, including realizability by linear torsion groups and sequential realizability, of which the latter is the most general. After showing that fusion systems of compact Lie groups are always realized by linear torsion groups (hence sequentially realizable), we give some new tools for showing that certain fusion systems are not sequentially realizable, and illustrate it with two large families of examples.
{"title":"Realizability of fusion systems by discrete groups","authors":"Carles Broto, Ran Levi, Bob Oliver","doi":"arxiv-2409.09703","DOIUrl":"https://doi.org/arxiv-2409.09703","url":null,"abstract":"For a prime $p$, fusion systems over discrete $p$-toral groups are categories\u0000that model and generalize the $p$-local structure of Lie groups and certain\u0000other infinite groups in the same way that fusion systems over finite\u0000$p$-groups model and generalize the $p$-local structure of finite groups. In\u0000the finite case, it is natural to say that a fusion system $mathcal{F}$ is\u0000realizable if it is isomorphic to the fusion system of a finite group, but it\u0000is less clear what realizability should mean in the discrete $p$-toral case. In this paper, we look at some of the different types of realizability for\u0000fusion systems over discrete $p$-toral groups, including realizability by\u0000linear torsion groups and sequential realizability, of which the latter is the\u0000most general. After showing that fusion systems of compact Lie groups are\u0000always realized by linear torsion groups (hence sequentially realizable), we\u0000give some new tools for showing that certain fusion systems are not\u0000sequentially realizable, and illustrate it with two large families of examples.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253757","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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