We explore the structure of invariant measures on compact Kähler manifolds with Hamiltonian torus actions. We derive the formula for conditional measures on the orbits of the complex torus and use it to prove a conditional statement about uniqueness of solutions to the g-Monge-Ampère equation.
{"title":"On the Conditional Measures on the Orbits of the Complex Torus","authors":"Szymon Myga","doi":"10.1307/mmj/20216050","DOIUrl":"https://doi.org/10.1307/mmj/20216050","url":null,"abstract":"We explore the structure of invariant measures on compact Kähler manifolds with Hamiltonian torus actions. We derive the formula for conditional measures on the orbits of the complex torus and use it to prove a conditional statement about uniqueness of solutions to the g-Monge-Ampère equation.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"65 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75809965","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}
Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) Hom(G,H) and the subfunctor of homomorphisms (of algebraic groups)Homgp(G,H). We show thatHom(G,H) is represented by a group scheme, locally of finite type, if the k-vector space O(G) is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that Homgp(G,H) is represented by a smooth scheme M ; moreover, every orbit of H acting by conjugation on M is open.
{"title":"Homomorphisms of Algebraic Groups: Representability and Rigidity","authors":"M. Brion","doi":"10.1307/mmj/20217214","DOIUrl":"https://doi.org/10.1307/mmj/20217214","url":null,"abstract":"Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) Hom(G,H) and the subfunctor of homomorphisms (of algebraic groups)Homgp(G,H). We show thatHom(G,H) is represented by a group scheme, locally of finite type, if the k-vector space O(G) is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that Homgp(G,H) is represented by a smooth scheme M ; moreover, every orbit of H acting by conjugation on M is open.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"15 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87496277","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, in a weakly regular p-adic Lie group G, the subgroup Gu spanned by the one-parameter subgroups of G admits a Levi decomposition. As a consequence, there exists a regular open subgroup of G which contains Gu
{"title":"How Far Are p-Adic Lie Groups from Algebraic Groups?","authors":"Y. Benoist, Jean-François Quint","doi":"10.1307/mmj/20217205","DOIUrl":"https://doi.org/10.1307/mmj/20217205","url":null,"abstract":"We show that, in a weakly regular p-adic Lie group G, the subgroup Gu spanned by the one-parameter subgroups of G admits a Levi decomposition. As a consequence, there exists a regular open subgroup of G which contains Gu","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80780462","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 prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.
{"title":"Limit Groups over Coherent Right-Angled Artin Groups Are Cyclic Subgroup Separable","authors":"J. Fruchter","doi":"10.1307/mmj/20216031","DOIUrl":"https://doi.org/10.1307/mmj/20216031","url":null,"abstract":"We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"29 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73483973","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 study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semi-explicit description of the locus of degree n+ 2 rational curves in Pn of arithmetic genus two that admit a Khovanskii-finite valuation. Furthermore, we describe an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, we show that rational curves with a single unibranch singularity are always Khovanskii-finite if their arithmetic genus is sufficiently small.
{"title":"Khovanskii-Finite Rational Curves of Arithmetic Genus 2","authors":"N. Ilten, Ahmad Mokhtar","doi":"10.1307/mmj/20216048","DOIUrl":"https://doi.org/10.1307/mmj/20216048","url":null,"abstract":"We study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semi-explicit description of the locus of degree n+ 2 rational curves in Pn of arithmetic genus two that admit a Khovanskii-finite valuation. Furthermore, we describe an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, we show that rational curves with a single unibranch singularity are always Khovanskii-finite if their arithmetic genus is sufficiently small.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"182 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80345974","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 study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the first examples of injective endomorphisms of mapping class groups (of surfaces with empty boundary) that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by homeomorphism. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.
{"title":"Big Mapping Class Groups and the Co-Hopfian Property","authors":"J. Aramayona, C. Leininger, A. McLeay","doi":"10.1307/mmj/20216075","DOIUrl":"https://doi.org/10.1307/mmj/20216075","url":null,"abstract":"We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the first examples of injective endomorphisms of mapping class groups (of surfaces with empty boundary) that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by homeomorphism. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"47 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77170291","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}
Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sub-linearly with v = μ(H/Γ), the co-volume of Γ in H. We prove this for non-uniform lattices in a very strong form, showing that for 2−generic such H’s, d(Γ) = OH(log v/ log log v), which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than d(Γ) = O(log v).
{"title":"On the Asymptotic Number of Generators of High Rank Arithmetic Lattices","authors":"A. Lubotzky, Raz Slutsky","doi":"10.1307/mmj/20217204","DOIUrl":"https://doi.org/10.1307/mmj/20217204","url":null,"abstract":"Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sub-linearly with v = μ(H/Γ), the co-volume of Γ in H. We prove this for non-uniform lattices in a very strong form, showing that for 2−generic such H’s, d(Γ) = OH(log v/ log log v), which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than d(Γ) = O(log v).","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"74 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85875434","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}
Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's"$mathfrak{sl}(n)$-like"Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $mathfrak{sl}(n)$ homology and $HFK_n$.
{"title":"Evaluations of Link Polynomials and Recent Constructions in Heegaard Floer Theory","authors":"Larry Gu, A. Manion","doi":"10.1307/mmj/20216061","DOIUrl":"https://doi.org/10.1307/mmj/20216061","url":null,"abstract":"Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's\"$mathfrak{sl}(n)$-like\"Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $mathfrak{sl}(n)$ homology and $HFK_n$.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72973148","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}
This paper compares notions of double sliceness for links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end we prove a result classifying Seifert fibered spaces which admit a smooth embeddings into integer homology $S^1 times S^3$s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links are also given.
{"title":"Doubly Slice Montesinos Links","authors":"D. McCoy, Clayton McDonald","doi":"10.1307/mmj/20216077","DOIUrl":"https://doi.org/10.1307/mmj/20216077","url":null,"abstract":"This paper compares notions of double sliceness for links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end we prove a result classifying Seifert fibered spaces which admit a smooth embeddings into integer homology $S^1 times S^3$s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links are also given.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"71 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85738132","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 analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree (n, 1) case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.
{"title":"Clark Measures for Rational Inner Functions","authors":"K. Bickel, J. Cima, A. Sola","doi":"10.1307/mmj/20216046","DOIUrl":"https://doi.org/10.1307/mmj/20216046","url":null,"abstract":"We analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree (n, 1) case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79249120","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}
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