Pub Date : 2022-10-01DOI: 10.1112/S0010437X22007734
Yi Ouyang, Jianfeng Xie
Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety $A$ over a global field $K$, the $p$-Selmer group $mathrm {Sel}_{p}(A/L)$ grows unboundedly when $L$ ranges over the $(mathbb {Z}/pmathbb {Z})$-extensions of $K$. Moreover, he raised a further problem: is $dim _{mathbb {F}_{p}} text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case $p neq mathrm {char},K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential $text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.
{"title":"The growth of Tate–Shafarevich groups in cyclic extensions","authors":"Yi Ouyang, Jianfeng Xie","doi":"10.1112/S0010437X22007734","DOIUrl":"https://doi.org/10.1112/S0010437X22007734","url":null,"abstract":"Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety $A$ over a global field $K$, the $p$-Selmer group $mathrm {Sel}_{p}(A/L)$ grows unboundedly when $L$ ranges over the $(mathbb {Z}/pmathbb {Z})$-extensions of $K$. Moreover, he raised a further problem: is $dim _{mathbb {F}_{p}} text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case $p neq mathrm {char},K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential $text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"158 1","pages":"2014 - 2032"},"PeriodicalIF":1.8,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41605227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-27DOI: 10.1112/S0010437X2300739X
Daniel Drimbe
We single out a large class of groups ${rm {boldsymbol {mathscr {M}}}}$ for which the following unique prime factorization result holds: if $Gamma _1,ldots,Gamma _nin {rm {boldsymbol {mathscr {M}}}}$ and $Gamma _1times cdots times Gamma _n$ is measure equivalent to a product $Lambda _1times cdots times Lambda _m$ of infinite icc groups, then $n ge m$, and if $n = m$, then, after permutation of the indices, $Gamma _i$ is measure equivalent to $Lambda _i$, for all $1leq ileq n$. This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to ${rm {boldsymbol {mathscr {M}}}}$. Class ${rm {boldsymbol {mathscr {M}}}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $Gamma$ for which either (i) $Gamma$ is an arbitrary wreath product group with amenable base or (ii) $Gamma$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${rm {boldsymbol {mathscr {M}}}}$. Finally, for groups $Gamma$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(Gamma )$ are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].
{"title":"Measure equivalence rigidity via s-malleable deformations","authors":"Daniel Drimbe","doi":"10.1112/S0010437X2300739X","DOIUrl":"https://doi.org/10.1112/S0010437X2300739X","url":null,"abstract":"We single out a large class of groups ${rm {boldsymbol {mathscr {M}}}}$ for which the following unique prime factorization result holds: if $Gamma _1,ldots,Gamma _nin {rm {boldsymbol {mathscr {M}}}}$ and $Gamma _1times cdots times Gamma _n$ is measure equivalent to a product $Lambda _1times cdots times Lambda _m$ of infinite icc groups, then $n ge m$, and if $n = m$, then, after permutation of the indices, $Gamma _i$ is measure equivalent to $Lambda _i$, for all $1leq ileq n$. This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to ${rm {boldsymbol {mathscr {M}}}}$. Class ${rm {boldsymbol {mathscr {M}}}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $Gamma$ for which either (i) $Gamma$ is an arbitrary wreath product group with amenable base or (ii) $Gamma$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${rm {boldsymbol {mathscr {M}}}}$. Finally, for groups $Gamma$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(Gamma )$ are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"159 1","pages":"2023 - 2050"},"PeriodicalIF":1.8,"publicationDate":"2022-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42122999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.1112/S0010437X23007364
A. Sah, Mehtaab Sawhney
Given a connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of $3,4,5$ edges as well as paths of $3$ edges in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $mathbb {G}(n,p)$.
{"title":"Subgraph distributions in dense random regular graphs","authors":"A. Sah, Mehtaab Sawhney","doi":"10.1112/S0010437X23007364","DOIUrl":"https://doi.org/10.1112/S0010437X23007364","url":null,"abstract":"Given a connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate manner on the occurrence and number of cycles of $3,4,5$ edges as well as paths of $3$ edges in $H$. More generally, we provide control of the asymptotic distribution of certain statistics of bounded degree which are invariant under vertex permutations, including moments of the spectrum of a random regular graph. Our techniques are based on combining complex-analytic methods due to McKay and Wormald used to enumerate regular graphs with the notion of graph factors developed by Janson in the context of studying subgraph counts in $mathbb {G}(n,p)$.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"159 1","pages":"2125 - 2148"},"PeriodicalIF":1.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45692798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.1112/S0010437X22007709
Yago Antolín, A. Jaikin-Zapirain
The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.
{"title":"The Hanna Neumann conjecture for surface groups","authors":"Yago Antolín, A. Jaikin-Zapirain","doi":"10.1112/S0010437X22007709","DOIUrl":"https://doi.org/10.1112/S0010437X22007709","url":null,"abstract":"The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"158 1","pages":"1850 - 1877"},"PeriodicalIF":1.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48371781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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