We construct a family of infinite simple groups that we call emph{twisted Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups $sV$ ($sinmathbb{N}$). We use twisted Brin-Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $sV$ and hence every right-angled Artin group, including examples of type $textrm{F}_infty$ and a family of examples of type $textrm{F}_{n-1}$ but not of type $textrm{F}_n$, for arbitrary $ninmathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.
{"title":"Twisted Brin–Thompson groups","authors":"James M. Belk, Matthew C. B. Zaremsky","doi":"10.2140/gt.2022.26.1189","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1189","url":null,"abstract":"We construct a family of infinite simple groups that we call emph{twisted Brin-Thompson groups}, generalizing Brin's higher-dimensional Thompson groups $sV$ ($sinmathbb{N}$). We use twisted Brin-Thompson groups to prove a variety of results regarding simple groups. For example, we prove that every finitely generated group embeds quasi-isometrically as a subgroup of a two-generated simple group, strengthening a result of Bridson. We also produce examples of simple groups that contain every $sV$ and hence every right-angled Artin group, including examples of type $textrm{F}_infty$ and a family of examples of type $textrm{F}_{n-1}$ but not of type $textrm{F}_n$, for arbitrary $ninmathbb{N}$. This provides the second known infinite family of simple groups distinguished by their finiteness properties.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132535554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Since its first use by Behrend, Bryan, and SzendrH{o}i in the computation of motivic Donaldson-Thomas (DT) invariants of $mathbb{A}_{mathbb{C}}^3$, dimensional reduction has proved to be an important tool in motivic and cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym, and SzendrH{o}i on motivic DT invariants, work of Dobrovolska, Ginzburg, and Travkin on exponential sums, and work of Orlov and Hirano on equivalences of categories of singularities, we generalize the dimensional reduction theorem in motivic and cohomological DT theory and use it to prove versions of the Cazzaniga-Morrison-Pym-SzendrH{o}i conjecture in these settings.
{"title":"Deformed dimensional reduction","authors":"Ben Davison, Tudor Puadurariu","doi":"10.2140/gt.2022.26.721","DOIUrl":"https://doi.org/10.2140/gt.2022.26.721","url":null,"abstract":"Since its first use by Behrend, Bryan, and SzendrH{o}i in the computation of motivic Donaldson-Thomas (DT) invariants of $mathbb{A}_{mathbb{C}}^3$, dimensional reduction has proved to be an important tool in motivic and cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym, and SzendrH{o}i on motivic DT invariants, work of Dobrovolska, Ginzburg, and Travkin on exponential sums, and work of Orlov and Hirano on equivalences of categories of singularities, we generalize the dimensional reduction theorem in motivic and cohomological DT theory and use it to prove versions of the Cazzaniga-Morrison-Pym-SzendrH{o}i conjecture in these settings.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126444497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We reconstruct the all-genus Fan-Jarvis-Ruan-Witten invariants of a Fermat cubic Landau-Ginzburg space $(x_1^3+x_2^3+x_3^3: [mathbb{C}^3/ mathbold{mu}_3]to mathbb{C})$ from genus-one primary invariants, using tautological relations and axioms of Cohomological Field Theories. These genus-one invariants satisfy a Chazy equation by the Belorousski-Pandharipande relation. They are completely determined by a single genus-one invariant, which can be obtained from cosection localization and intersection theory on moduli of three spin curves. We solve an all-genus Landau-Ginzburg/Calabi-Yau Correspondence Conjecture for the Fermat cubic Landau-Ginzburg space using Cayley transformation on quasi-modular forms. This transformation relates two non-semisimple CohFT theories: the Fan-Jarvis-Ruan-Witten theory of the Fermat cubic polynomial and the Gromov-Witten theory of the Fermat cubic curve. As a consequence, Fan-Jarvis-Ruan-Witten invariants at any genus can be computed using Gromov-Witten invariants of the elliptic curve. They also satisfy nice structures including holomorphic anomaly equations and Virasoro constraints.
{"title":"Higher genus FJRW invariants of a Fermat cubic","authors":"Jun Li, Yefeng Shen, Jie Zhou","doi":"10.2140/gt.2023.27.1845","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1845","url":null,"abstract":"We reconstruct the all-genus Fan-Jarvis-Ruan-Witten invariants of a Fermat cubic Landau-Ginzburg space $(x_1^3+x_2^3+x_3^3: [mathbb{C}^3/ mathbold{mu}_3]to mathbb{C})$ from genus-one primary invariants, using tautological relations and axioms of Cohomological Field Theories. These genus-one invariants satisfy a Chazy equation by the Belorousski-Pandharipande relation. They are completely determined by a single genus-one invariant, which can be obtained from cosection localization and intersection theory on moduli of three spin curves. We solve an all-genus Landau-Ginzburg/Calabi-Yau Correspondence Conjecture for the Fermat cubic Landau-Ginzburg space using Cayley transformation on quasi-modular forms. This transformation relates two non-semisimple CohFT theories: the Fan-Jarvis-Ruan-Witten theory of the Fermat cubic polynomial and the Gromov-Witten theory of the Fermat cubic curve. As a consequence, Fan-Jarvis-Ruan-Witten invariants at any genus can be computed using Gromov-Witten invariants of the elliptic curve. They also satisfy nice structures including holomorphic anomaly equations and Virasoro constraints.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126767876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding $S^1$-equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (non-equivariant) open-closed map is. These {em cyclic open-closed maps} give (a) constructions of geometric smooth and/or proper Calabi-Yau structures on Fukaya categories (which in the proper case implies the Fukaya category has a cyclic A-infinity model in characteristic 0) and (b) a purely symplectic proof of the non-commutative Hodge-de Rham degeneration conjecture for smooth and proper subcategories of Fukaya categories of compact symplectic manifolds. Further applications of cyclic open-closed maps, to counting curves in mirror symmetry and to comparing topological field theories, are the subject of joint projects with Perutz-Sheridan [GPS1, GPS2] and Cohen [CG].
{"title":"Cyclic homology, S1–equivariant Floer\u0000cohomology and Calabi–Yau structures","authors":"Sheel Ganatra","doi":"10.2140/gt.2023.27.3461","DOIUrl":"https://doi.org/10.2140/gt.2023.27.3461","url":null,"abstract":"We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding $S^1$-equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (non-equivariant) open-closed map is. These {em cyclic open-closed maps} give (a) constructions of geometric smooth and/or proper Calabi-Yau structures on Fukaya categories (which in the proper case implies the Fukaya category has a cyclic A-infinity model in characteristic 0) and (b) a purely symplectic proof of the non-commutative Hodge-de Rham degeneration conjecture for smooth and proper subcategories of Fukaya categories of compact symplectic manifolds. Further applications of cyclic open-closed maps, to counting curves in mirror symmetry and to comparing topological field theories, are the subject of joint projects with Perutz-Sheridan [GPS1, GPS2] and Cohen [CG].","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"87 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138600185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping class groups have local coarse boundedness (the analog of local compactness). When the end space of the surface is countable or tame, we also give a classification of those surface where there exists a coarsely bounded generating set (the analog of finite or compact generation, giving the group a well-defined quasi-isometry type) and those surfaces with mapping class groups of bounded diameter (the analog of compactness).
{"title":"Large-scale geometry of big mapping class groups","authors":"Kathryn Mann, Kasra Rafi","doi":"10.2140/gt.2023.27.2237","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2237","url":null,"abstract":"We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping class groups have local coarse boundedness (the analog of local compactness). When the end space of the surface is countable or tame, we also give a classification of those surface where there exists a coarsely bounded generating set (the analog of finite or compact generation, giving the group a well-defined quasi-isometry type) and those surfaces with mapping class groups of bounded diameter (the analog of compactness).","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"152 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114617580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperkahler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.
{"title":"Stability conditions and moduli spaces for\u0000Kuznetsov components of Gushel–Mukai varieties","authors":"Alexander Perry, L. Pertusi, Xiaolei Zhao","doi":"10.2140/gt.2022.26.3055","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3055","url":null,"abstract":"We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperkahler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128972983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A conjecture of Morel asserts that the sheaf of $mathbb A^1$-connected components of a space is $mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $mathbb A^1$-connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel's conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.
{"title":"𝔸1–connected components of ruled\u0000surfaces","authors":"Chetan T. Balwe, Anand Sawant","doi":"10.2140/gt.2022.26.321","DOIUrl":"https://doi.org/10.2140/gt.2022.26.321","url":null,"abstract":"A conjecture of Morel asserts that the sheaf of $mathbb A^1$-connected components of a space is $mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $mathbb A^1$-connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel's conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117126169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly.
{"title":"The spheres of Sol","authors":"Matei P. Coiculescu, R. Schwartz","doi":"10.2140/gt.2022.26.2103","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2103","url":null,"abstract":"Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133488107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider discrete groups in ${rm PGL}_d(mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be relatively hyperbolic in terms of the geometry of the convex domain. This answers a question of Danciger-Gu'eritaud-Kassel and is analogous to a result of Hruska-Kleiner for ${rm CAT}(0)$ spaces.
{"title":"Convex cocompact actions of relatively hyperbolic groups","authors":"Mitul Islam, Andrew M. Zimmer","doi":"10.2140/gt.2023.27.417","DOIUrl":"https://doi.org/10.2140/gt.2023.27.417","url":null,"abstract":"In this paper we consider discrete groups in ${rm PGL}_d(mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be relatively hyperbolic in terms of the geometry of the convex domain. This answers a question of Danciger-Gu'eritaud-Kassel and is analogous to a result of Hruska-Kleiner for ${rm CAT}(0)$ spaces.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124961216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $S$ be a closed surface of genus $g geq 2$ and let $rho$ be a maximal $mathrm{PSL}(2, mathbb{R}) times mathrm{PSL}(2, mathbb{R})$ surface group representation. By a result of Schoen, there is a unique $rho$-equivariant minimal surface $widetilde{Sigma}$ in $mathbb{H}^{2} times mathbb{H}^{2}$. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. In the second half of the paper, we provide a geometric interpretation: the minimal surfaces $widetilde{Sigma}$ degenerate to the core of a product of two $mathbb{R}$-trees. As a consequence, we obtain a compactification of the space of maximal representations of $pi_{1}(S)$ into $mathrm{PSL}(2, mathbb{R}) times mathrm{PSL}(2, mathbb{R})$.
设$S$为属$g geq 2$的一个封闭曲面，设$rho$为一个极大的$mathrm{PSL}(2, mathbb{R}) times mathrm{PSL}(2, mathbb{R})$曲面群表示。根据Schoen的结果，在$mathbb{H}^{2} times mathbb{H}^{2}$中存在唯一的$rho$ -等变最小曲面$widetilde{Sigma}$。我们研究了这些极小曲面上的诱导度量，并证明了它们的极限是精确混合结构。在论文的第二部分，我们提供了一个几何解释:最小曲面$widetilde{Sigma}$退化到两个$mathbb{R}$ -树的乘积的核心。因此，我们得到了$pi_{1}(S)$的极大表示空间的紧化到$mathrm{PSL}(2, mathbb{R}) times mathrm{PSL}(2, mathbb{R})$。
{"title":"High-energy harmonic maps and degeneration of minimal surfaces","authors":"Charles Ouyang","doi":"10.2140/gt.2023.27.1691","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1691","url":null,"abstract":"Let $S$ be a closed surface of genus $g geq 2$ and let $rho$ be a maximal $mathrm{PSL}(2, mathbb{R}) times mathrm{PSL}(2, mathbb{R})$ surface group representation. By a result of Schoen, there is a unique $rho$-equivariant minimal surface $widetilde{Sigma}$ in $mathbb{H}^{2} times mathbb{H}^{2}$. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. In the second half of the paper, we provide a geometric interpretation: the minimal surfaces $widetilde{Sigma}$ degenerate to the core of a product of two $mathbb{R}$-trees. As a consequence, we obtain a compactification of the space of maximal representations of $pi_{1}(S)$ into $mathrm{PSL}(2, mathbb{R}) times mathrm{PSL}(2, mathbb{R})$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128644426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：