. We generalize Bestvina’s notion of a Z -boundary for a group to that of a “coarse Z -boundary.” We show that established theorems about Z -boundaries carry over nicely to the more general theory, and that some wished-for properties of Z -boundaries become theorems when applied to coarse Z -boundaries. Most notably, the property of admitting a coarse Z -boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by in-troducing the notion of a “model Z -geometry.” In accordance with the existing theory, we also develop an equivariant version of the above—that of a “coarse E Z -boundary.”
{"title":"Coarse Z-Boundaries for Groups","authors":"C. Guilbault, Molly A. Moran","doi":"10.1307/mmj/20206001","DOIUrl":"https://doi.org/10.1307/mmj/20206001","url":null,"abstract":". We generalize Bestvina’s notion of a Z -boundary for a group to that of a “coarse Z -boundary.” We show that established theorems about Z -boundaries carry over nicely to the more general theory, and that some wished-for properties of Z -boundaries become theorems when applied to coarse Z -boundaries. Most notably, the property of admitting a coarse Z -boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by in-troducing the notion of a “model Z -geometry.” In accordance with the existing theory, we also develop an equivariant version of the above—that of a “coarse E Z -boundary.”","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"20 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72466181","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}
{"title":"Pseudo-Néron Model and Restriction of Sections","authors":"Santai Qu","doi":"10.1307/mmj/20195764","DOIUrl":"https://doi.org/10.1307/mmj/20195764","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90769718","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}
{"title":"A Note on the Concordance Z-Genus","authors":"Allison N. Miller, Junghwan Park","doi":"10.1307/mmj/20216070","DOIUrl":"https://doi.org/10.1307/mmj/20216070","url":null,"abstract":"","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"2005 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88341389","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}
. It’s well-known that adding a general boundary would create K-stability. As an application, we reprove product theorem for delta invariants of Fano varieties.
。众所周知，添加一般边界会产生k稳定性。作为应用，我们重新证明了Fano变量不变量的乘积定理。
{"title":"Product Theorem on Delta Invariants via Adding a General Boundary","authors":"Chuyu Zhou","doi":"10.1307/mmj/20205993","DOIUrl":"https://doi.org/10.1307/mmj/20205993","url":null,"abstract":". It’s well-known that adding a general boundary would create K-stability. As an application, we reprove product theorem for delta invariants of Fano varieties.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"83 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79649187","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 arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:1408.4708, and that it is therefore not well-defined.
{"title":"The dlt Motivic Zeta Function Is Not Well Defined","authors":"J. Nicaise, Naud Potemans, W. Veys","doi":"10.1307/mmj/20216148","DOIUrl":"https://doi.org/10.1307/mmj/20216148","url":null,"abstract":"In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:1408.4708, and that it is therefore not well-defined.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"37 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72752589","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. In the late seventies, Sullivan showed that for a convex cocompact subgroup Γ of SO(n, 1) with critical exponent δ > 0, any Γ-conformal measure on ∂H of dimension δ is necessarily supported on the limit set Λ and that the conformal measure of dimension δ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup Γ of a connected semisimple real algebraic group G of rank at most 3. We also obtain the local mixing for generalized BMS measures on ΓG including Haar measures.
{"title":"Uniqueness of Conformal Measures and Local Mixing for Anosov Groups","authors":"Sam O. Edwards, Minju M. Lee, H. Oh","doi":"10.1307/mmj/20217222","DOIUrl":"https://doi.org/10.1307/mmj/20217222","url":null,"abstract":"Abstract. In the late seventies, Sullivan showed that for a convex cocompact subgroup Γ of SO(n, 1) with critical exponent δ > 0, any Γ-conformal measure on ∂H of dimension δ is necessarily supported on the limit set Λ and that the conformal measure of dimension δ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup Γ of a connected semisimple real algebraic group G of rank at most 3. We also obtain the local mixing for generalized BMS measures on ΓG including Haar measures.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"183 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73273864","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 use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.
我们用g签名定理来定义类似于结签名的强可逆结的不变量。
{"title":"Strongly Invertible Knots, Invariant Surfaces, and the Atiyah–Singer Signature Theorem","authors":"Antonio Alfieri, Keegan Boyle","doi":"10.1307/mmj/20226183","DOIUrl":"https://doi.org/10.1307/mmj/20226183","url":null,"abstract":"We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"44 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86121659","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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