Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.
{"title":"An Intrinsic Characterization of Bruhat–Tits Buildings Inside Analytic Groups","authors":"Bertrand R'emy, Amaury Thuillier, A. Werner","doi":"10.1307/mmj/20217220","DOIUrl":"https://doi.org/10.1307/mmj/20217220","url":null,"abstract":"Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"48 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81165073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J.-L. Colliot-Th'elene, D. Harbater, Julia Hartmann, D. Krashen, R. Parimala, V. Suresh
We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.
{"title":"Local-Global Principles for Constant Reductive Groups over Semi-Global Fields","authors":"J.-L. Colliot-Th'elene, D. Harbater, Julia Hartmann, D. Krashen, R. Parimala, V. Suresh","doi":"10.1307/mmj/20217219","DOIUrl":"https://doi.org/10.1307/mmj/20217219","url":null,"abstract":"We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; i.e., over one variable function fields over complete discretely valued fields. We provide conditions on the group and the semiglobal field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"103 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74879133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. We characterize all determinant values coprime to $p$, give sharp divisibility conditions for multiples of $p$, and determine all values when $p=3$. We also provide new sharp conditions on the power of $p$ dividing the group determinants for $mathbb Z_p^2$. For a finite group, the integer group determinants can be understood as corresponding to Lind's generalization of the Mahler measure. We speculate on the Lind-Mahler measure for the discrete Heisenberg group and for two other infinite non-abelian groups arising from symmetries of the plane and 3-space.
{"title":"The Integer Group Determinants for the Heisenberg Group of Order p3","authors":"Michael J. Mossinghoff, Christopher G. Pinner","doi":"10.1307/mmj/20216124","DOIUrl":"https://doi.org/10.1307/mmj/20216124","url":null,"abstract":"We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. We characterize all determinant values coprime to $p$, give sharp divisibility conditions for multiples of $p$, and determine all values when $p=3$. We also provide new sharp conditions on the power of $p$ dividing the group determinants for $mathbb Z_p^2$. For a finite group, the integer group determinants can be understood as corresponding to Lind's generalization of the Mahler measure. We speculate on the Lind-Mahler measure for the discrete Heisenberg group and for two other infinite non-abelian groups arising from symmetries of the plane and 3-space.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"5 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83034165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Γ be a non-elementary Kleinian group and H < Γ a finitely generated, proper subgroup. We prove that if Γ has finite co-volume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3) and of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C). Dedicated to Gopal Prasad on the occasion of his 75th birthday
{"title":"Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property","authors":"M. Bridson, A. Reid","doi":"10.1307/mmj/20217218","DOIUrl":"https://doi.org/10.1307/mmj/20217218","url":null,"abstract":"Let Γ be a non-elementary Kleinian group and H < Γ a finitely generated, proper subgroup. We prove that if Γ has finite co-volume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3) and of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C). Dedicated to Gopal Prasad on the occasion of his 75th birthday","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"16 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88773215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eduardo Mart'inez-Pedroza, Luis Jorge S'anchez Saldana
For a finitely generated group $G$, let $H(G)$ denote Bowditch's taut loop length spectrum. We prove that if $G=(Aast B) / langle!langle mathcal R rangle!rangle $ is a $C'(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $H(G)$ is equivalent to $H(A) cup H(B)$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch's construction of continuously many non-quasi-isometric $C'(1/6)$ small cancellation $2$-generated groups to obtain our main result: Let $mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: $bulletleft{Gin mathcal{G} colon underline{mathrm{cd}}(G) = 2 text{ and } underline{mathrm{gd}}(G) = 3 right}$ $bulletleft{Gin mathcal{G} colon underline{underline{mathrm{cd}}}(G) = 2 text{ and } underline{underline{mathrm{gd}}}(G) = 3 right}$ $bulletleft{Gin mathcal{G} colon mathrm{cd}_{mathbb{Q}}(G)=2 text{ and } mathrm{cd}_{mathbb{Z}}(G)=3 right}$ On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented $C'(1/12)$ small cancellation quotients of free products, in particular, this produces new examples of groups exhibiting an Eilenberg-Ganea phenomenon for families. We also show that if there is a finitely presented counter-example to the Eilenberg-Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counter-examples.
对于有限生成的群$G$，设$H(G)$表示Bowditch的紧环长度谱。证明了如果$G=(Aast B) / langle!langle mathcal R rangle!rangle $是有限生成群的自由积的一个$C'(1/12)$小消商，则$H(G)$等价于$H(A) cup H(B)$。我们将这一结果与上同维和几何维的界以及Bowditch构造的连续许多非拟等距$C'(1/6)$小消去$2$生成群结合起来，得到了我们的主要结果:设$mathcal{G}$表示有限生成群的类别。下面的子类包含连续的许多单端非拟等长群:$bulletleft{Gin mathcal{G} colon underline{mathrm{cd}}(G) = 2 text{ and } underline{mathrm{gd}}(G) = 3 right}$$bulletleft{Gin mathcal{G} colon underline{underline{mathrm{cd}}}(G) = 2 text{ and } underline{underline{mathrm{gd}}}(G) = 3 right}$$bulletleft{Gin mathcal{G} colon mathrm{cd}_{mathbb{Q}}(G)=2 text{ and } mathrm{cd}_{mathbb{Z}}(G)=3 right}$在我们证明上述结果的过程中，我们证明了上面定义的类在相对有限的情况下是封闭的$C'(1/12)$小的自由积的消商，特别是，这产生了显示家庭的Eilenberg-Ganea现象的群的新例子。我们还证明了如果存在一个有限生成的Eilenberg-Ganea猜想的反例，那么就存在连续多个有限生成的单端非拟等距反例。
{"title":"Bowditch Taut Spectrum and Dimensions of Groups","authors":"Eduardo Mart'inez-Pedroza, Luis Jorge S'anchez Saldana","doi":"10.1307/mmj/20216121","DOIUrl":"https://doi.org/10.1307/mmj/20216121","url":null,"abstract":"For a finitely generated group $G$, let $H(G)$ denote Bowditch's taut loop length spectrum. We prove that if $G=(Aast B) / langle!langle mathcal R rangle!rangle $ is a $C'(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $H(G)$ is equivalent to $H(A) cup H(B)$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch's construction of continuously many non-quasi-isometric $C'(1/6)$ small cancellation $2$-generated groups to obtain our main result: Let $mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: $bulletleft{Gin mathcal{G} colon underline{mathrm{cd}}(G) = 2 text{ and } underline{mathrm{gd}}(G) = 3 right}$ $bulletleft{Gin mathcal{G} colon underline{underline{mathrm{cd}}}(G) = 2 text{ and } underline{underline{mathrm{gd}}}(G) = 3 right}$ $bulletleft{Gin mathcal{G} colon mathrm{cd}_{mathbb{Q}}(G)=2 text{ and } mathrm{cd}_{mathbb{Z}}(G)=3 right}$ On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented $C'(1/12)$ small cancellation quotients of free products, in particular, this produces new examples of groups exhibiting an Eilenberg-Ganea phenomenon for families. We also show that if there is a finitely presented counter-example to the Eilenberg-Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counter-examples.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91273930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.
{"title":"The Minkowski Equality of Big Divisors","authors":"S. Cutkosky","doi":"10.1307/mmj/20216107","DOIUrl":"https://doi.org/10.1307/mmj/20216107","url":null,"abstract":"We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"7 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74198108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the Ham Sandwich theorem and (iv) the Lyusternik--Shnirel'man--Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.
我们证明了射影空间中码的大小控制着从球到欧氏空间的奇映射的零的结构结果。实际上，这种关系是通过球面上的概率测度空间的拓扑来给出的，球面的支承直径有特定的参数限定。我们的主要成果是对Borsuk—Ulam定理的推广，并得到了它的四个结果:(1)给出了simmonyi和Tardos关于图的圆色数拓扑下界的一个新的证明;(ii)研究了球面在欧几里得空间中的一般嵌入，并证明了投影码给出了球面嵌入的一般测度的定量界;并且我们证明了(iii) Ham Sandwich定理和(iv) Lyusternik—Shnirel’man—Borsuk覆盖定理的推广，分别适用于覆盖中的测度数或集合数可能超过环境维数的情况。
{"title":"The Topology of Projective Codes and the Distribution of Zeros of Odd Maps","authors":"Henry Adams, Johnathan Bush, F. Frick","doi":"10.1307/mmj/20216170","DOIUrl":"https://doi.org/10.1307/mmj/20216170","url":null,"abstract":"We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the Ham Sandwich theorem and (iv) the Lyusternik--Shnirel'man--Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"46 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88789500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Katie Gittins, Carolyn Gordon, Magda Khalile, I. M. Solis, Mary R. Sandoval, E. Stanhope
We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $le 3.$
{"title":"Do the Hodge Spectra Distinguish Orbifolds from Manifolds? Part 1","authors":"Katie Gittins, Carolyn Gordon, Magda Khalile, I. M. Solis, Mary R. Sandoval, E. Stanhope","doi":"10.1307/mmj/20216126","DOIUrl":"https://doi.org/10.1307/mmj/20216126","url":null,"abstract":"We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $le 3.$","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"182 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77374887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Friedman--Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).
{"title":"An Analogue of the Strengthened Hanna Neumann Conjecture for Virtually Free Groups and Virtually Free Products","authors":"A. Klyachko, A. Zakharov","doi":"10.1307/mmj/20216105","DOIUrl":"https://doi.org/10.1307/mmj/20216105","url":null,"abstract":"The Friedman--Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"95 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73625014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora, but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces, which recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.
{"title":"Khovanov Homology for Links in Thickened Multipunctured Disks","authors":"Zachary Winkeler","doi":"10.1307/mmj/20216166","DOIUrl":"https://doi.org/10.1307/mmj/20216166","url":null,"abstract":"We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora, but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces, which recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"76 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76270457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：