Pub Date : 2021-01-24DOI: 10.1142/s1793525321500606
David Munoz, Jorge Plazas, Mario Vel'asquez
In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.
{"title":"Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups","authors":"David Munoz, Jorge Plazas, Mario Vel'asquez","doi":"10.1142/s1793525321500606","DOIUrl":"https://doi.org/10.1142/s1793525321500606","url":null,"abstract":"In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"113 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80649722","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}
Pub Date : 2021-01-01DOI: 10.1142/s1793525321500461
Michael Brannan, Li Gao, M. Junge
We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.
{"title":"Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II","authors":"Michael Brannan, Li Gao, M. Junge","doi":"10.1142/s1793525321500461","DOIUrl":"https://doi.org/10.1142/s1793525321500461","url":null,"abstract":"We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90284322","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}
Pub Date : 2020-12-31DOI: 10.1142/s1793525321500503
Matthew Stover
The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.
{"title":"Geometry of the Wiman–Edge monodromy","authors":"Matthew Stover","doi":"10.1142/s1793525321500503","DOIUrl":"https://doi.org/10.1142/s1793525321500503","url":null,"abstract":"The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"67 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72458500","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}
Pub Date : 2020-12-11DOI: 10.1142/s1793525321500151
J. Roydor
We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any von Neumann algebra which is close enough is actually Jordan ∗-isomorphic. These vanishing conditions are possibly empty.
{"title":"Banach–Mazur stability of von Neumann algebras","authors":"J. Roydor","doi":"10.1142/s1793525321500151","DOIUrl":"https://doi.org/10.1142/s1793525321500151","url":null,"abstract":"We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any von Neumann algebra which is close enough is actually Jordan ∗-isomorphic. These vanishing conditions are possibly empty.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"8 1","pages":"1-26"},"PeriodicalIF":0.8,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78727238","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}
Pub Date : 2020-12-09DOI: 10.1142/s1793525321500187
Joanna Markowicz, S. Prus
An estimate for the modulus of convexity and characteristic of convexity of a general direct sum of Banach spaces is established. Using direct sums, we construct a space with given characteristic of convexity and the value of the modulus of convexity at [Formula: see text]. A result on uniform convexity of spaces obtained with the general discrete interpolation method is proved.
{"title":"Uniform convexity of general direct sums and interpolation spaces","authors":"Joanna Markowicz, S. Prus","doi":"10.1142/s1793525321500187","DOIUrl":"https://doi.org/10.1142/s1793525321500187","url":null,"abstract":"An estimate for the modulus of convexity and characteristic of convexity of a general direct sum of Banach spaces is established. Using direct sums, we construct a space with given characteristic of convexity and the value of the modulus of convexity at [Formula: see text]. A result on uniform convexity of spaces obtained with the general discrete interpolation method is proved.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89536167","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}
Pub Date : 2020-12-01DOI: 10.1142/S1793525320500028
D. Burago, Jin Lu, Tristan Ozuch
Given a diffeomorphism which is homotopic to the identity from the [Formula: see text]-torus to itself, we construct an isotopy whose norm is controlled by that of the diffeomorphism in question.
{"title":"How large isotopy is needed to connect homotopic diffeomorphisms (of 𝕋2)","authors":"D. Burago, Jin Lu, Tristan Ozuch","doi":"10.1142/S1793525320500028","DOIUrl":"https://doi.org/10.1142/S1793525320500028","url":null,"abstract":"Given a diffeomorphism which is homotopic to the identity from the [Formula: see text]-torus to itself, we construct an isotopy whose norm is controlled by that of the diffeomorphism in question.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77136994","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}
Pub Date : 2020-11-21DOI: 10.1142/s1793525321500096
Assaf Bar-Natan, M. Duchin, Robert P. Kropholler
We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as “medium-scale” because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature [Formula: see text], abelian groups are identically flat, and in the other direction we show that [Formula: see text] implies the group is virtually abelian. Beyond that, [Formula: see text] captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study.
{"title":"Conjugation curvature for Cayley graphs","authors":"Assaf Bar-Natan, M. Duchin, Robert P. Kropholler","doi":"10.1142/s1793525321500096","DOIUrl":"https://doi.org/10.1142/s1793525321500096","url":null,"abstract":"We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as “medium-scale” because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature [Formula: see text], abelian groups are identically flat, and in the other direction we show that [Formula: see text] implies the group is virtually abelian. Beyond that, [Formula: see text] captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"158 1","pages":"1-21"},"PeriodicalIF":0.8,"publicationDate":"2020-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74853824","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}
Pub Date : 2020-11-19DOI: 10.1142/S1793525323500048
Jake Solomon, Sara B. Tukachinsky
We give a detailed account of differential forms and currents on orbifolds with corners, the pull-back and push-forward operations, and their fundamental properties. We work within the formalism where the category of orbifolds with corners is obtained as a localization of the category of etale proper groupoids with corners. Constructions and proofs are formulated in terms of the structure maps of the groupoids, avoiding the use of orbifold charts. The Frechet space of differential forms on an orbifold and the dual space of currents are shown to be independent of which etale proper groupoid is chosen to represent the orbifold.
{"title":"Differential forms on orbifolds with corners","authors":"Jake Solomon, Sara B. Tukachinsky","doi":"10.1142/S1793525323500048","DOIUrl":"https://doi.org/10.1142/S1793525323500048","url":null,"abstract":"We give a detailed account of differential forms and currents on orbifolds with corners, the pull-back and push-forward operations, and their fundamental properties. We work within the formalism where the category of orbifolds with corners is obtained as a localization of the category of etale proper groupoids with corners. Constructions and proofs are formulated in terms of the structure maps of the groupoids, avoiding the use of orbifold charts. The Frechet space of differential forms on an orbifold and the dual space of currents are shown to be independent of which etale proper groupoid is chosen to represent the orbifold.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82108869","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}
Pub Date : 2020-11-16DOI: 10.1142/s1793525321500072
Shengkui Ye
Let [Formula: see text] be the special linear group over integers and [Formula: see text] [Formula: see text], or [Formula: see text] products of spheres and tori. We prove that any group action of [Formula: see text] on [Formula: see text] by diffeomorphims or piecewise linear homeomorphisms is trivial if [Formula: see text] This confirms a conjecture on Zimmer’s program for these manifolds.
{"title":"Matrix group actions on product of spheres","authors":"Shengkui Ye","doi":"10.1142/s1793525321500072","DOIUrl":"https://doi.org/10.1142/s1793525321500072","url":null,"abstract":"Let [Formula: see text] be the special linear group over integers and [Formula: see text] [Formula: see text], or [Formula: see text] products of spheres and tori. We prove that any group action of [Formula: see text] on [Formula: see text] by diffeomorphims or piecewise linear homeomorphisms is trivial if [Formula: see text] This confirms a conjecture on Zimmer’s program for these manifolds.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"6 1","pages":"1-19"},"PeriodicalIF":0.8,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83614988","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}
Pub Date : 2020-11-12DOI: 10.1142/s1793525321500138
Rita Gitik
We define a system of groups, a duality system of groups and a PD(n)-system of groups, generalizing the corresponding concepts of pairs of groups. We give several characterizations of duality syste...
{"title":"Duality systems of groups and PD(n)-systems of groups","authors":"Rita Gitik","doi":"10.1142/s1793525321500138","DOIUrl":"https://doi.org/10.1142/s1793525321500138","url":null,"abstract":"We define a system of groups, a duality system of groups and a PD(n)-system of groups, generalizing the corresponding concepts of pairs of groups. We give several characterizations of duality syste...","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"45 1","pages":"1-12"},"PeriodicalIF":0.8,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75222900","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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