In this article, we provide bounds on systoles associated to a holomorphic $1$-form $omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.
{"title":"The maximum number of systoles for genus two Riemann surfaces with abelian differentials","authors":"C. Judge, H. Parlier","doi":"10.4171/CMH/463","DOIUrl":"https://doi.org/10.4171/CMH/463","url":null,"abstract":"In this article, we provide bounds on systoles associated to a holomorphic $1$-form $omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/463","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44315036","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}
Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.
{"title":"Generalized triangle groups, expanders, and a problem of Agol and Wise","authors":"A. Lubotzky, J. Manning, H. Wilton","doi":"10.17863/CAM.27372","DOIUrl":"https://doi.org/10.17863/CAM.27372","url":null,"abstract":"Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49427426","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 $F_n$ be the free group on $n$ generators and $Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklos Abert.
{"title":"Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups","authors":"Michael Brandenbursky, Michał Marcinkowski","doi":"10.4171/cmh/470","DOIUrl":"https://doi.org/10.4171/cmh/470","url":null,"abstract":"Let $F_n$ be the free group on $n$ generators and $Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklos Abert.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/470","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49095817","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 determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10.
{"title":"Essential dimension of the spin groups in characteristic 2","authors":"B. Totaro","doi":"10.4171/CMH/452","DOIUrl":"https://doi.org/10.4171/CMH/452","url":null,"abstract":"We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. \u0000We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/452","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46223302","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 $G$ be a semisimple Lie group and $Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $Gammabackslash G$.
{"title":"On the rate of equidistribution of expanding translates of horospheres in $Gammabackslash G$","authors":"Samuel C. Edwards","doi":"10.4171/cmh/513","DOIUrl":"https://doi.org/10.4171/cmh/513","url":null,"abstract":"Let $G$ be a semisimple Lie group and $Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $Gammabackslash G$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44232039","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}
For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincare-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological "rotational" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.
{"title":"A Poincaré–Bendixson theorem for translation lines and applications to prime ends","authors":"Andres Koropecki, A. Passeggi","doi":"10.4171/CMH/457","DOIUrl":"https://doi.org/10.4171/CMH/457","url":null,"abstract":"For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincare-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological \"rotational\" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/457","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42648664","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 moduli space QMg of non-zero genus g quadratic differentials has a natural action of G D GL 2 .R/= ̋ ̇ 1 0 0 1 ̨ . The Veech group PSL.X; q/ is the stabilizer of .X; q/ 2 QMg in G. We describe a new algorithm for finding elements of PSL.X; q/ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H=PSL.X; q/ is greater than zero. Mathematics Subject Classification (2010). 32G15, 30F30.
{"title":"Fundamental domains and generators for lattice Veech groups","authors":"R. E. Mukamel","doi":"10.4171/CMH/406","DOIUrl":"https://doi.org/10.4171/CMH/406","url":null,"abstract":"The moduli space QMg of non-zero genus g quadratic differentials has a natural action of G D GL 2 .R/= ̋ ̇ 1 0 0 1 ̨ . The Veech group PSL.X; q/ is the stabilizer of .X; q/ 2 QMg in G. We describe a new algorithm for finding elements of PSL.X; q/ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H=PSL.X; q/ is greater than zero. Mathematics Subject Classification (2010). 32G15, 30F30.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"92 1","pages":"57-83"},"PeriodicalIF":0.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/406","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70840626","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}
P. F. D. Santos, R. Hardt, James Lewis, P. Lima-filho
{"title":"An explicit cycle map for the motivic cohomology of real varieties","authors":"P. F. D. Santos, R. Hardt, James Lewis, P. Lima-filho","doi":"10.4171/CMH/416","DOIUrl":"https://doi.org/10.4171/CMH/416","url":null,"abstract":"","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"92 1","pages":"429-465"},"PeriodicalIF":0.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/416","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70840764","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 construct connections on $S^1$-equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters.
我们构造了$S^1$-等变哈密顿花上同调上的连接，这些连接对某些形式参数微分。
{"title":"Connections on equivariant Hamiltonian Floer cohomology","authors":"P. Seidel","doi":"10.4171/CMH/445","DOIUrl":"https://doi.org/10.4171/CMH/445","url":null,"abstract":"We construct connections on $S^1$-equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/445","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841563","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}
Camillo De Lellis, Andrea Marchese, E. Spadaro, Daniele Valtorta
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.
{"title":"Rectifiability and upper Minkowski bounds for singularities of harmonic $Q$-valued maps","authors":"Camillo De Lellis, Andrea Marchese, E. Spadaro, Daniele Valtorta","doi":"10.4171/CMH/449","DOIUrl":"https://doi.org/10.4171/CMH/449","url":null,"abstract":"In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/449","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841972","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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