We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.
{"title":"Self-Similar Surfaces: Involutions and Perfection","authors":"Justin Malestein, Jing Tao","doi":"10.1307/mmj/20216114","DOIUrl":"https://doi.org/10.1307/mmj/20216114","url":null,"abstract":"We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"14 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74466271","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}
Abstract. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (with A fixed but arbitrary). We also show that the values of s(n), for n composite, are uniformly distributed among all p residue classes modulo every p ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.
{"title":"Distribution mod p of Euler’s Totient and the Sum of Proper Divisors","authors":"Noah Lebowitz-Lockard, P. Pollack, A. Roy","doi":"10.1307/mmj/20216082","DOIUrl":"https://doi.org/10.1307/mmj/20216082","url":null,"abstract":"Abstract. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (with A fixed but arbitrary). We also show that the values of s(n), for n composite, are uniformly distributed among all p residue classes modulo every p ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73345891","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 a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $dim V/G
{"title":"Generic Stabilizers for Simple Algebraic Groups","authors":"S. Garibaldi, R. Guralnick","doi":"10.1307/mmj/20217216","DOIUrl":"https://doi.org/10.1307/mmj/20217216","url":null,"abstract":"We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $dim V/G<dim G$, or there is a $v in V$ whose stabilizer in $G$ is trivial.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"74 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86354346","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 estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.
{"title":"Weyl Sums over Integers with Digital Restrictions","authors":"I. Shparlinski, J. Thuswaldner","doi":"10.1307/mmj/20216094","DOIUrl":"https://doi.org/10.1307/mmj/20216094","url":null,"abstract":"We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"85 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84339093","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 abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich--Rafi and Dowdall--Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.
{"title":"Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups","authors":"Carolyn R. Abbott, J. Manning","doi":"10.1307/mmj/20216112","DOIUrl":"https://doi.org/10.1307/mmj/20216112","url":null,"abstract":"We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich--Rafi and Dowdall--Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"105 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80868968","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 maximal multiplicity locus of a variety X over a field of characteristic p > 0 that is provided with a finite surjective radicial morphism δ : X → V , where V is regular, for example, when X ⊂ A is a hypersurface defined by an equation of the form T −f(x1, . . . , xn) = 0 and δ is the projection onto V := Spec(k[x1, . . . , xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V ) ⊂ K(X). We denote by Fd(X) ⊂ X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities x ∈ Fd(X) with a good behavior property under blowups X → X along regular centers included in Fd(X), which we call invariants with the pointwise inequality property. A finite radicial morphism δ : X → V as above will be expressed in terms of an O V -submodule M ⊆ OV . A blowup X → X along a regular equimultiple center included in Fd(X) induces a blowup V ′ → V along a regular center and a finite morphism δ : X → V . A notion of transform of the O V -module M ⊂ OV to an O V ′ -module M ′ ⊂ OV ′ will be defined in such a way that δ ′ : X → V ′ is the radicial morphism defined by M . Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on O V -submodules M ⊂ OV .
{"title":"Multiplicity Along Points of a Radicial Covering of a Regular Variety","authors":"Diego Sulca, O. Villamayor","doi":"10.1307/MMJ/20195775","DOIUrl":"https://doi.org/10.1307/MMJ/20195775","url":null,"abstract":"We study the maximal multiplicity locus of a variety X over a field of characteristic p > 0 that is provided with a finite surjective radicial morphism δ : X → V , where V is regular, for example, when X ⊂ A is a hypersurface defined by an equation of the form T −f(x1, . . . , xn) = 0 and δ is the projection onto V := Spec(k[x1, . . . , xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V ) ⊂ K(X). We denote by Fd(X) ⊂ X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities x ∈ Fd(X) with a good behavior property under blowups X → X along regular centers included in Fd(X), which we call invariants with the pointwise inequality property. A finite radicial morphism δ : X → V as above will be expressed in terms of an O V -submodule M ⊆ OV . A blowup X → X along a regular equimultiple center included in Fd(X) induces a blowup V ′ → V along a regular center and a finite morphism δ : X → V . A notion of transform of the O V -module M ⊂ OV to an O V ′ -module M ′ ⊂ OV ′ will be defined in such a way that δ ′ : X → V ′ is the radicial morphism defined by M . Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on O V -submodules M ⊂ OV .","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"39 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82553534","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 an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $beta r^delta$ for some explicit constants $beta, delta>0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.
{"title":"Effective Discreteness Radius of Stabilizers for Stationary Actions","authors":"T. Gelander, Arie Levit, G. Margulis","doi":"10.1307/mmj/20217209","DOIUrl":"https://doi.org/10.1307/mmj/20217209","url":null,"abstract":"We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $beta r^delta$ for some explicit constants $beta, delta>0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"5 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88688339","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 K be a compact convex domain in the Euclidean plane. The mixed area A(K,−K) of K and−K can be bounded from above by 1/(6 √ 3)L(K)2, where L(K) is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil (1991). They also showed that if K is a polygon, then equality holds if and only if K is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality 6 √ 3A(K,−K) ≤ L(K)2.
{"title":"Extremizers and Stability of the Betke–Weil Inequality","authors":"F. Bartha, Ferenc Bencs, K. Boroczky, D. Hug","doi":"10.1307/mmj/20216063","DOIUrl":"https://doi.org/10.1307/mmj/20216063","url":null,"abstract":"Let K be a compact convex domain in the Euclidean plane. The mixed area A(K,−K) of K and−K can be bounded from above by 1/(6 √ 3)L(K)2, where L(K) is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil (1991). They also showed that if K is a polygon, then equality holds if and only if K is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality 6 √ 3A(K,−K) ≤ L(K)2.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"17 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76577628","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}
In 1979 Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T) $ P $ C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Z f̂(n) ξn exp(2πint) in the general setting of dependent random variables (ξn).
1979年，Pisier显著地证明了一个独立的同分布的标准高斯随机变量序列，通过随机傅立叶级数，决定了严格包含在C(T)中的齐次巴纳赫代数P，严格包含经典维纳代数a (T)的单位圆T上的连续函数类，即a (T) $ P $ C(T)。这改进了Zafran在解决卡兹尼尔森提出的一个长期存在的问题时获得的一些先前的结果。本文推广了Pisier的结果，证明了单位圆上的任何概率测度都定义了C(T)中包含的齐次Banach代数。因此，皮西耶代数不是一个孤立的对象，而是一个大的皮西耶代数类的元素。考虑高斯随机变量平稳序列谱测度的情况，得到了随机傅里叶级数∑n∈Z f²(n) ξn exp(2πint)在相依随机变量(ξn)一般设置下有界性的一个充分条件。
{"title":"A Generalization of Pisier Homogeneous Banach Algebra","authors":"Safari Mukeru","doi":"10.1307/mmj/20205914","DOIUrl":"https://doi.org/10.1307/mmj/20205914","url":null,"abstract":"In 1979 Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T) $ P $ C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Z f̂(n) ξn exp(2πint) in the general setting of dependent random variables (ξn).","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"88 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74036838","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 a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite {'e}tale cover.
{"title":"When Is a Reductive Group Scheme Linear?","authors":"P. Gille","doi":"10.1307/mmj/20217208","DOIUrl":"https://doi.org/10.1307/mmj/20217208","url":null,"abstract":"We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite {'e}tale cover.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"67 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73392976","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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