A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $mathbf{PSO}_{7,1}(mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that no arithmetic lattice in $mathbf{PO}_{n,1}(mathbb{R})$, $n>3$, is LERF.
{"title":"Arithmetic Trialitarian Hyperbolic Lattices Are Not Locally Extended Residually Finite","authors":"Nikolay Bogachev, Leone Slavich, Hongbin Sun","doi":"10.1093/imrn/rnae053","DOIUrl":"https://doi.org/10.1093/imrn/rnae053","url":null,"abstract":"A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $mathbf{PSO}_{7,1}(mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that no arithmetic lattice in $mathbf{PO}_{n,1}(mathbb{R})$, $n>3$, is LERF.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"7 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that for any two Riemannian metrics $sigma _{1}, sigma _{2}$ on the unit disk, a homeomorphism $partial mathbb{D}to partial mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(mathbb{D},sigma _{1})to (mathbb{D},sigma _{2})$ with $L^{1}$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^{1}$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees.
{"title":"Minimal Diffeomorphisms with L1 Hopf Differentials","authors":"Nathaniel Sagman","doi":"10.1093/imrn/rnae049","DOIUrl":"https://doi.org/10.1093/imrn/rnae049","url":null,"abstract":"We prove that for any two Riemannian metrics $sigma _{1}, sigma _{2}$ on the unit disk, a homeomorphism $partial mathbb{D}to partial mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(mathbb{D},sigma _{1})to (mathbb{D},sigma _{2})$ with $L^{1}$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^{1}$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"30 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss a notion of discrete conformal equivalence for decorated piecewise Euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. Under the assumption that the circles are non-intersecting, we prove the corresponding discrete uniformization theorem. The uniformization theorem for discrete conformal maps corresponds to the special case that all circles degenerate to points. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. It is based on a concave variational principle, which also provides a method for the computation of decorated discrete conformal maps.
我们讨论了装饰性片状欧几里得曲面（PE-surface）的离散共形等价性概念，即每个顶点可选择一个圆的 PE-surface。它与反向距离和超理想圆模式密切相关。在圆不相交的假设下，我们证明了相应的离散均匀化定理。离散共形映射的均匀化定理对应于所有圆退化为点的特殊情况。我们的证明依赖于装饰 PE 曲面、双曲曲面的典型细分曲面和凸双曲多面体之间的密切关系。它以凹变分原理为基础，同时也提供了计算装饰离散保角映射的方法。
{"title":"Decorated Discrete Conformal Maps and Convex Polyhedral Cusps","authors":"Alexander I Bobenko, Carl O R Lutz","doi":"10.1093/imrn/rnae016","DOIUrl":"https://doi.org/10.1093/imrn/rnae016","url":null,"abstract":"We discuss a notion of discrete conformal equivalence for decorated piecewise Euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. Under the assumption that the circles are non-intersecting, we prove the corresponding discrete uniformization theorem. The uniformization theorem for discrete conformal maps corresponds to the special case that all circles degenerate to points. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. It is based on a concave variational principle, which also provides a method for the computation of decorated discrete conformal maps.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"VII 1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a notion of a point-wise entropy of measures (i.e., local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension.
{"title":"Neutralized Local Entropy and Dimension bounds for Invariant Measures","authors":"S Ben Ovadia, F Rodriguez-Hertz","doi":"10.1093/imrn/rnae047","DOIUrl":"https://doi.org/10.1093/imrn/rnae047","url":null,"abstract":"We introduce a notion of a point-wise entropy of measures (i.e., local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"44 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $C$ be a smooth projective curve over ${{mathbb{C}}}$. We link the periodicity of Hitchin’s uniformizing Higgs bundle of $C$ with the underlying arithmetic geometry of the curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.
{"title":"Periodicity of Hitchin’s Uniformizing Higgs Bundles","authors":"Raju Krishnamoorthy, Mao Sheng","doi":"10.1093/imrn/rnae042","DOIUrl":"https://doi.org/10.1093/imrn/rnae042","url":null,"abstract":"Let $C$ be a smooth projective curve over ${{mathbb{C}}}$. We link the periodicity of Hitchin’s uniformizing Higgs bundle of $C$ with the underlying arithmetic geometry of the curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"161 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a description of a certain induced module for a quantum group of type $A$. Together with our previous results this gives a proof of Lusztig’s conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type quantized enveloping algebra of type $A_{n}$ at the $ell $-th root of unity, where $ell $ is an odd integer satisfying $(ell ,n+1)=1$ and $ell> n+1$.
{"title":"The Koszul Complex and a Certain Induced Module for a Quantum group","authors":"Toshiyuki Tanisaki","doi":"10.1093/imrn/rnae043","DOIUrl":"https://doi.org/10.1093/imrn/rnae043","url":null,"abstract":"We give a description of a certain induced module for a quantum group of type $A$. Together with our previous results this gives a proof of Lusztig’s conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type quantized enveloping algebra of type $A_{n}$ at the $ell $-th root of unity, where $ell $ is an odd integer satisfying $(ell ,n+1)=1$ and $ell> n+1$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that every bounded domain in a metric measure space can be approximated in measure from inside by closed $BV$-extension sets. The extension sets are obtained by minimizing the sum of the perimeter and the measure of the difference between the domain and the set. By earlier results, in PI spaces the minimizers have open representatives with locally quasiminimal surface. We give an example in a PI space showing that the open representative of the minimizer need not be a $BV$-extension domain nor locally John.
我们证明，公度量空间中的每个有界域都可以用封闭的 $BV$ 扩展集从内部逼近度量。扩展集是通过最小化域与集之间的周长之和与差的度量而得到的。根据早先的结果，在 PI 空间中，最小化集具有局部准最小曲面的开放代表。我们给出了一个 PI 空间的例子，说明最小化的开放代表不一定是 $BV$ 扩展域，也不一定是局部约翰。
{"title":"Approximation by BV-extension Sets via Perimeter Minimization in Metric Spaces","authors":"Jesse Koivu, Danka Lučić, Tapio Rajala","doi":"10.1093/imrn/rnae048","DOIUrl":"https://doi.org/10.1093/imrn/rnae048","url":null,"abstract":"We show that every bounded domain in a metric measure space can be approximated in measure from inside by closed $BV$-extension sets. The extension sets are obtained by minimizing the sum of the perimeter and the measure of the difference between the domain and the set. By earlier results, in PI spaces the minimizers have open representatives with locally quasiminimal surface. We give an example in a PI space showing that the open representative of the minimizer need not be a $BV$-extension domain nor locally John.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $C$ be a unipotent class of $G=textrm{SO}(N,mathbb{C})$, $mathcal{E}$ an irreducible $G$-equivariant local system on $C$. The generalized Springer representation $rho (C,mathcal{E})$ appears in the top cohomology of some variety. Let $bar rho (C,mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well known that $rho (C,mathcal{E})$ appears in $bar rho (C,mathcal{E})$ with multiplicity $1$ and that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $bar rho (C,mathcal{E})$. Suppose $C$ is parametrized by an orthogonal partition with only odd parts. We prove that $bar rho (C,mathcal{E})$ (resp. $textrm{sgn}otimes bar rho (C,mathcal{E})$) has a unique multiplicity 1 “maximal” subrepresentation $rho ^{textrm{max}}$ (resp. “minimal” subrepresentation $textrm{sgn}otimes rho ^{textrm{max}}$), where $textrm{sgn}$ is the sign representation. These are analogues of results for $textrm{Sp}(2n,mathbb{C})$ by Waldspurger.
{"title":"Maximality Properties of Generalized Springer Representations of SO (N, ℂ)","authors":"Ruben La","doi":"10.1093/imrn/rnae041","DOIUrl":"https://doi.org/10.1093/imrn/rnae041","url":null,"abstract":"Let $C$ be a unipotent class of $G=textrm{SO}(N,mathbb{C})$, $mathcal{E}$ an irreducible $G$-equivariant local system on $C$. The generalized Springer representation $rho (C,mathcal{E})$ appears in the top cohomology of some variety. Let $bar rho (C,mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well known that $rho (C,mathcal{E})$ appears in $bar rho (C,mathcal{E})$ with multiplicity $1$ and that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $bar rho (C,mathcal{E})$. Suppose $C$ is parametrized by an orthogonal partition with only odd parts. We prove that $bar rho (C,mathcal{E})$ (resp. $textrm{sgn}otimes bar rho (C,mathcal{E})$) has a unique multiplicity 1 “maximal” subrepresentation $rho ^{textrm{max}}$ (resp. “minimal” subrepresentation $textrm{sgn}otimes rho ^{textrm{max}}$), where $textrm{sgn}$ is the sign representation. These are analogues of results for $textrm{Sp}(2n,mathbb{C})$ by Waldspurger.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"20 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $(textrm{SL}_{2}(mathbb{R}))^{n} rtimes S_{n}$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $mathbb{F}$, there is no finite set of equations whose orbit under $(textrm{SL}_{2}(mathbb{F}))^{n} rtimes S_{n}$ cuts out the image of $ntimes n$ matrices over ${mathbb{F}}$ under the principal minor map for every $n$.
{"title":"Determinantal Representations and the Image of the Principal Minor Map","authors":"Abeer Al Ahmadieh, Cynthia Vinzant","doi":"10.1093/imrn/rnae038","DOIUrl":"https://doi.org/10.1093/imrn/rnae038","url":null,"abstract":"In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $(textrm{SL}_{2}(mathbb{R}))^{n} rtimes S_{n}$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $mathbb{F}$, there is no finite set of equations whose orbit under $(textrm{SL}_{2}(mathbb{F}))^{n} rtimes S_{n}$ cuts out the image of $ntimes n$ matrices over ${mathbb{F}}$ under the principal minor map for every $n$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"162 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.
{"title":"Linkage and F-Regularity of Determinantal Rings","authors":"Vaibhav Pandey, Yevgeniya Tarasova","doi":"10.1093/imrn/rnae040","DOIUrl":"https://doi.org/10.1093/imrn/rnae040","url":null,"abstract":"In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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