{"title":"Zariski dense surface groups in\u0000SL(2k + 1, ℤ)","authors":"D. D. Long, M. Thistlethwaite","doi":"10.2140/gt.2024.28.1153","DOIUrl":"https://doi.org/10.2140/gt.2024.28.1153","url":null,"abstract":"","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":" 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140992426","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}
{"title":"Correction to the article Bimodules in bordered Heegaard Floer homology","authors":"Robert Lipshitz, P. Ozsváth, D. Thurston","doi":"10.2140/gt.2024.28.1001","DOIUrl":"https://doi.org/10.2140/gt.2024.28.1001","url":null,"abstract":"","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"2003 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140246503","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}
{"title":"The nonabelian Brill–Noether divisor on ℳ13\u0000and the Kodaira dimension of ℛ13","authors":"Gavril Farkas, David Jensen, Sam Payne","doi":"10.2140/gt.2024.28.803","DOIUrl":"https://doi.org/10.2140/gt.2024.28.803","url":null,"abstract":"","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"275 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140247314","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 show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for cubulated hyperbolic groups, and applies to a wide range of peripheral subgroups. In particular, we deduce virtual specialness for properly and cocompactly cubulated groups that are hyperbolic relative to virtually abelian groups. As another consequence, by using a theorem of Martin and Steenbock we obtain virtual specialness for groups obtained as a quotient of a free product of finitely many virtually compact special groups by a finite set of relators satisfying the classical $C'(1/6)$-small cancellation condition.
{"title":"On cubulated relatively hyperbolic groups","authors":"E. Reyes","doi":"10.2140/gt.2023.27.575","DOIUrl":"https://doi.org/10.2140/gt.2023.27.575","url":null,"abstract":"We show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for cubulated hyperbolic groups, and applies to a wide range of peripheral subgroups. In particular, we deduce virtual specialness for properly and cocompactly cubulated groups that are hyperbolic relative to virtually abelian groups. As another consequence, by using a theorem of Martin and Steenbock we obtain virtual specialness for groups obtained as a quotient of a free product of finitely many virtually compact special groups by a finite set of relators satisfying the classical $C'(1/6)$-small cancellation condition.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"132 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128214317","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}
Democritus and the early atomists held that "the material cause of all things that exist is the coming together of atoms and void. Atoms are eternal and have many different shapes, and they can cluster together to create things that are perceivable. Differences in shape, arrangement, and position of atoms produce different phenomena". Like the atoms of Democritus, the Grim Reaper solution to curve shortening flow is eternal and indivisible -- it does not split off a line, and is itself its only "asymptotic translator". Confirming the heuristic described by Huisken and Sinestrari [J. Differential Geom. 101, 2 (2015), 267-287], we show that it gives rise to a great diversity of convex ancient and translating solutions to mean curvature flow, through the evolution of families of Grim hyperplanes in suitable configurations. We construct, in all dimensions $nge 2$, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. The latter resolve a conjecture of White [J. Amer. Math. Soc. 16, 1 (2003), 123-138]. We also provide a detailed asymptotic analysis of convex ancient solutions in slab regions in general. Roughly speaking, we show that they decompose "backwards in time" into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. An analogous decomposition holds "forwards in time" for eternal solutions. One consequence is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the mid-plane of their slab.
{"title":"Ancient mean curvature flows out of polytopes","authors":"T. Bourni, Mathew T. Langford, G. Tinaglia","doi":"10.2140/gt.2022.26.1849","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1849","url":null,"abstract":"Democritus and the early atomists held that \"the material cause of all things that exist is the coming together of atoms and void. Atoms are eternal and have many different shapes, and they can cluster together to create things that are perceivable. Differences in shape, arrangement, and position of atoms produce different phenomena\". Like the atoms of Democritus, the Grim Reaper solution to curve shortening flow is eternal and indivisible -- it does not split off a line, and is itself its only \"asymptotic translator\". Confirming the heuristic described by Huisken and Sinestrari [J. Differential Geom. 101, 2 (2015), 267-287], we show that it gives rise to a great diversity of convex ancient and translating solutions to mean curvature flow, through the evolution of families of Grim hyperplanes in suitable configurations. We construct, in all dimensions $nge 2$, a large family of new examples, including both symmetric and asymmetric examples, as well as many eternal examples that do not evolve by translation. The latter resolve a conjecture of White [J. Amer. Math. Soc. 16, 1 (2003), 123-138]. We also provide a detailed asymptotic analysis of convex ancient solutions in slab regions in general. Roughly speaking, we show that they decompose \"backwards in time\" into a canonical configuration of Grim hyperplanes which satisfies certain necessary conditions. An analogous decomposition holds \"forwards in time\" for eternal solutions. One consequence is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the mid-plane of their slab.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"260 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123692611","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 work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way a natural interpolation, realized in practice by the existence of a natural filtration, from Hochschild and cyclic homology to derived de Rham cohomology. The construction our filtered circle is based upon the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.
{"title":"A universal Hochschild–Kostant–Rosenberg\u0000theorem","authors":"Tasos Moulinos, Marco Robalo, B. Toën","doi":"10.2140/gt.2022.26.777","DOIUrl":"https://doi.org/10.2140/gt.2022.26.777","url":null,"abstract":"In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way a natural interpolation, realized in practice by the existence of a natural filtration, from Hochschild and cyclic homology to derived de Rham cohomology. The construction our filtered circle is based upon the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117155926","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 show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture of Margalit. Moreover we recover results of Artin and Lin about the classification of homomorphisms from braid groups on n strands to symmetric groups on k letters, where k is at most n. Unlike the original proofs, our method does not use the Bertrand-Chebyshev theorem, answering a question of Artin. Similarly for mapping class group of closed orientable surfaces, the smallest non-cyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary proof of this result, originally due to Kielak-Pierro, which proves a conjecture of Zimmermann.
{"title":"Smallest noncyclic quotients of braid and mapping class groups","authors":"S. Kolay","doi":"10.2140/gt.2023.27.2479","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2479","url":null,"abstract":"We show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture of Margalit. Moreover we recover results of Artin and Lin about the classification of homomorphisms from braid groups on n strands to symmetric groups on k letters, where k is at most n. Unlike the original proofs, our method does not use the Bertrand-Chebyshev theorem, answering a question of Artin. Similarly for mapping class group of closed orientable surfaces, the smallest non-cyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary proof of this result, originally due to Kielak-Pierro, which proves a conjecture of Zimmermann.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133130566","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 X be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on X leading to bounds on the counting function in Geometric Manin’s Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin’s Conjecture for certain split del Pezzo surfaces of degree ≥ 2 admitting a birational morphism to P over the ground field.
{"title":"Classifying sections of del Pezzo fibrations, II","authors":"Brian Lehmann, Sho Tanimoto","doi":"10.2140/gt.2022.26.2565","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2565","url":null,"abstract":"Let X be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on X leading to bounds on the counting function in Geometric Manin’s Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin’s Conjecture for certain split del Pezzo surfaces of degree ≥ 2 admitting a birational morphism to P over the ground field.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126965454","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}
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