In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
{"title":"Boundary rigidity of negatively-curved asymptotically hyperbolic surfaces","authors":"Thibault Lefeuvre","doi":"10.4171/cmh/483","DOIUrl":"https://doi.org/10.4171/cmh/483","url":null,"abstract":"In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/483","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41886755","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 provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than $sqrt{2}$ is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.
{"title":"Normal generators for mapping class groups are abundant","authors":"Justin Lanier, D. Margalit","doi":"10.4171/cmh/526","DOIUrl":"https://doi.org/10.4171/cmh/526","url":null,"abstract":"We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than $sqrt{2}$ is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47097277","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}
M. Duchin, V. Erlandsson, C. Leininger, C. Sadanand
We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.
{"title":"You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces","authors":"M. Duchin, V. Erlandsson, C. Leininger, C. Sadanand","doi":"10.4171/cmh/516","DOIUrl":"https://doi.org/10.4171/cmh/516","url":null,"abstract":"We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41617396","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}
Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},dots,p_{r}$ there exists a continuous probability measure $mu $ on the unit circle $mathbb{T}$ such that [ inf_{k_{1}ge 0,dots,k_{r}ge 0}|widehat{mu }(p_{1}^{k_{1}}dots p_{r}^{k_{r}})|>0. ] This results applies in particular to the Furstenberg set $F={2^{k}3^{k'},;,kge 0, k'ge 0}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $times 2$-$times 3$ conjecture. We also estimate the modified Kazhdan constant of $F$ and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.
{"title":"Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences","authors":"C. Badea, S. Grivaux","doi":"10.4171/cmh/482","DOIUrl":"https://doi.org/10.4171/cmh/482","url":null,"abstract":"Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},dots,p_{r}$ there exists a continuous probability measure $mu $ on the unit circle $mathbb{T}$ such that [ inf_{k_{1}ge 0,dots,k_{r}ge 0}|widehat{mu }(p_{1}^{k_{1}}dots p_{r}^{k_{r}})|>0. ] This results applies in particular to the Furstenberg set $F={2^{k}3^{k'},;,kge 0, k'ge 0}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $times 2$-$times 3$ conjecture. We also estimate the modified Kazhdan constant of $F$ and obtain general results on rigidity sequences which allow us to retrieve essentially all known examples of such sequences.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/482","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47812092","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 field, $A$ a finitely generated associative $k$-algebra and $operatorname{Rep}_A[n]$ the functor $operatorname{Fields}_kto operatorname{Sets}$, which sends a field $K$ containing $k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n) := operatorname{ed}_k(operatorname{Rep}_A[n])$, as $ntoinfty$. In particular, we show that the rate of growth of $r_A(n)$ determines the representation type of $A$. That is, $r_A(n)$ is bounded from above if $A$ is of finite representation type, grows linearly if $A$ is of tame representation type and grows quadratically if A is of wild representation type. Moreover, $r_A(n)$ is a finer invariant of A, which allows us to distinguish among algebras of the same representation type.
{"title":"Essential dimension of representations of algebras","authors":"F. Scavia","doi":"10.4171/cmh/500","DOIUrl":"https://doi.org/10.4171/cmh/500","url":null,"abstract":"Let $k$ be a field, $A$ a finitely generated associative $k$-algebra and $operatorname{Rep}_A[n]$ the functor $operatorname{Fields}_kto operatorname{Sets}$, which sends a field $K$ containing $k$ to the set of isomorphism classes of representations of $A_K$ of dimension at most $n$. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function $r_A(n) := operatorname{ed}_k(operatorname{Rep}_A[n])$, as $ntoinfty$. In particular, we show that the rate of growth of $r_A(n)$ determines the representation type of $A$. That is, $r_A(n)$ is bounded from above if $A$ is of finite representation type, grows linearly if $A$ is of tame representation type and grows quadratically if A is of wild representation type. Moreover, $r_A(n)$ is a finer invariant of A, which allows us to distinguish among algebras of the same representation type.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45563001","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 extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${mathbb R}^{n+p}$ is singularly genuinely rigid in ${mathbb R}^{n+q}$, for any $q < min{5,n} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.
{"title":"Singular genuine rigidity","authors":"L. Florit, Felippe Guimarães","doi":"10.4171/cmh/488","DOIUrl":"https://doi.org/10.4171/cmh/488","url":null,"abstract":"We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${mathbb R}^{n+p}$ is singularly genuinely rigid in ${mathbb R}^{n+q}$, for any $q < min{5,n} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/488","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46256434","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 classify simple groups that act by birational transformations on compact complex K"ahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.
{"title":"Simple groups of birational transformations in dimension two","authors":"Christian Urech","doi":"10.4171/cmh/486","DOIUrl":"https://doi.org/10.4171/cmh/486","url":null,"abstract":"We classify simple groups that act by birational transformations on compact complex K\"ahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/486","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48094379","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 the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.
{"title":"Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero","authors":"Olivier Benoist, J. C. Ottem","doi":"10.4171/cmh/479","DOIUrl":"https://doi.org/10.4171/cmh/479","url":null,"abstract":"We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/479","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49334720","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 this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.
{"title":"Characterization of generic projective space bundles and algebraicity of foliations","authors":"Carolina Araujo, S. Druel","doi":"10.4171/cmh/475","DOIUrl":"https://doi.org/10.4171/cmh/475","url":null,"abstract":"In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/475","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45429026","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 a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane.
{"title":"Rigidity of Busemann convex Finsler metrics","authors":"S. Ivanov, A. Lytchak","doi":"10.4171/cmh/476","DOIUrl":"https://doi.org/10.4171/cmh/476","url":null,"abstract":"We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/476","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43105482","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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