We exhibit explicit infinite families of finitely presented, Kazhdan, simple groups that are pairwise not measure equivalent. These groups are lattices acting on products of buildings. We obtain the result by studying vanishing and non-vanishing of their $L^2$-Betti numbers.
{"title":"Finitely presented simple groups and measure equivalence","authors":"Antonio L'opez Neumann","doi":"10.4064/cm8610-11-2022","DOIUrl":"https://doi.org/10.4064/cm8610-11-2022","url":null,"abstract":"We exhibit explicit infinite families of finitely presented, Kazhdan, simple groups that are pairwise not measure equivalent. These groups are lattices acting on products of buildings. We obtain the result by studying vanishing and non-vanishing of their $L^2$-Betti numbers.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44163550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 number of zeros of harmonic polynomials with complex coefficients. We also study the zeros of harmonic polynomials with complex coefficients which admit a decomposition F ( z ) = P n ( z ) + Q m ( z ) where P n and Q m are polynomials of degree n and m (respectively), with n > m . 30C10, 30C15, 42A50.
． 我们最高的墙当家》研究调和定律和情结coe polynomialsfficients。我们也研究调和定律之墙和情结coe polynomialsfficients哪种承认a decomposition F (z) = P (z) + n Q n m (z)在P和Q是学位中的polynomials n和m (respectively)里,用n > m。30C10 30C15 42A50
{"title":"Zeros of harmonic polynomials with\u0000complex coefficients","authors":"Chahrazed Harrat","doi":"10.4064/cm7874-4-2021","DOIUrl":"https://doi.org/10.4064/cm7874-4-2021","url":null,"abstract":". We study the maximal number of zeros of harmonic polynomials with complex coefficients. We also study the zeros of harmonic polynomials with complex coefficients which admit a decomposition F ( z ) = P n ( z ) + Q m ( z ) where P n and Q m are polynomials of degree n and m (respectively), with n > m . 30C10, 30C15, 42A50.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70139886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some topological and metric properties of\u0000the space of $ell _1$-predual hyperplanes in $c$","authors":"Agnieszka Gergont, Łukasz Piasecki","doi":"10.4064/cm8561-7-2021","DOIUrl":"https://doi.org/10.4064/cm8561-7-2021","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70140988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Corrigendum to “On the conjecture of Ulam on the invariance of measure in the Hilbert cube” (Colloq. Math. 152 (2018), 79–95)","authors":"Soon-Mo Jung, E. C. Kim","doi":"10.4064/cm7145c-12-2018","DOIUrl":"https://doi.org/10.4064/cm7145c-12-2018","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"164 1","pages":"161-170"},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70116339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an integral inequality for compact orientable real hypersurfaces of the complex quadric Q (n ≥ 3) in terms of their shape operator S and Reeb vector field ξ. As direct consequences, we obtain new characterizations for real hypersurfaces of Q with isometric Reeb flow. Such hypersurfaces have been classified by J. Berndt and Y. J. Suh [Int. J. Math. 24 (2013), art. 1350050, 18 pp.].
利用复二次曲面Q (n≥3)的形状算子S和Reeb向量场ξ证明了其紧致可定向实超曲面的一个积分不等式。作为直接结果，我们得到了具有等距Reeb流的Q实超曲面的新特征。这样的超曲面已经被J. Berndt和Y. J. Suh分类[j]。数学学报，24(2013)，第1期。[1350050, 18页]。
{"title":"New characterizations of real hypersurfaces with isometric Reeb flow in the complex quadric","authors":"Zejun Hu, Jiabin Yin","doi":"10.4064/cm8075-12-2019","DOIUrl":"https://doi.org/10.4064/cm8075-12-2019","url":null,"abstract":"We prove an integral inequality for compact orientable real hypersurfaces of the complex quadric Q (n ≥ 3) in terms of their shape operator S and Reeb vector field ξ. As direct consequences, we obtain new characterizations for real hypersurfaces of Q with isometric Reeb flow. Such hypersurfaces have been classified by J. Berndt and Y. J. Suh [Int. J. Math. 24 (2013), art. 1350050, 18 pp.].","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70133981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 generalized tilting module over the endomorphism algebra of an Oppermann–Thomas cluster tilting object in a 2n-Calabi–Yau (n + 2)-angulated category lifts to a relative cluster tilting object in this category. As an application, this generalizes a recent work of Fu and Liu for triangulated categories.
{"title":"Lifting to relative cluster tilting objects in $2n$-Calabi–Yau $(n+2)$-angulated categories","authors":"Panyue Zhou, Xing-fei Zhou","doi":"10.4064/cm8178-7-2020","DOIUrl":"https://doi.org/10.4064/cm8178-7-2020","url":null,"abstract":"We show that a generalized tilting module over the endomorphism algebra of an Oppermann–Thomas cluster tilting object in a 2n-Calabi–Yau (n + 2)-angulated category lifts to a relative cluster tilting object in this category. As an application, this generalizes a recent work of Fu and Liu for triangulated categories.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"125 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70135724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We show that the measure of maximal entropy for the hereditary closure of a B -free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, we show that B is taut whenever the corresponding Mirsky measure ν η has full support. This is the converse to a recent result of Keller.
{"title":"Hereditary subshifts whose measure of maximal entropy does not have the Gibbs property","authors":"J. Kułaga-Przymus, Michał Lemańczyk","doi":"10.4064/CM8223-11-2020","DOIUrl":"https://doi.org/10.4064/CM8223-11-2020","url":null,"abstract":". We show that the measure of maximal entropy for the hereditary closure of a B -free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, we show that B is taut whenever the corresponding Mirsky measure ν η has full support. This is the converse to a recent result of Keller.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70136782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We study a class of elliptic inequalities which arise in the study of blow-up rate estimates for parabolic problems, and obtain a sharp existence/nonexistence result. Namely, for any p ≥ 1 , we show that the inequality ∆u + u p ≤ ε in R n with u (0) = 1 admits a radial, positive nonincreasing solution for all ε > 0 if and only if n ≥ 2 . This solves a problem left open in [Souplet & Tayachi, Colloq. Math. 88 (2001)]. The result stands in contrast with the classical case ε = 0 .
{"title":"Sharp condition for the Liouville property in a class of nonlinear elliptic inequalities","authors":"P. Souplet","doi":"10.4064/cm8147-1-2020","DOIUrl":"https://doi.org/10.4064/cm8147-1-2020","url":null,"abstract":". We study a class of elliptic inequalities which arise in the study of blow-up rate estimates for parabolic problems, and obtain a sharp existence/nonexistence result. Namely, for any p ≥ 1 , we show that the inequality ∆u + u p ≤ ε in R n with u (0) = 1 admits a radial, positive nonincreasing solution for all ε > 0 if and only if n ≥ 2 . This solves a problem left open in [Souplet & Tayachi, Colloq. Math. 88 (2001)]. The result stands in contrast with the classical case ε = 0 .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70135440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ergodic properties of convolution operators in group algebras","authors":"H. Mustafayev, H. Topal","doi":"10.4064/CM8214-6-2020","DOIUrl":"https://doi.org/10.4064/CM8214-6-2020","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70135996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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