Hyperbolicity and dismantlability

双曲性和可拆卸性

• 批准号: RGPIN-2014-05409
• 负责人: Przytycki, Piotr
• 金额: $ 2.31万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635421
• 关键词: Hyperbolicity; dismantlability

The program consists of studying geometric topology and group theory questions as hyperbolicity or the fixed-point theorem using combinatorial methods with the prominent role of a graph-theoretic property called dismantlability. The long-term objectives are:I. Find more classes of hyperbolic groups that are residually finite.II. Apply dismantlability in geometric topology. I. Examples of hyperbolic groups are groups acting geometrically on the hyperbolic space, small cancellation groups, random groups, and high dimensional 7-systolic groups constructed by Haglund and Januszkiewicz-Swiatkowski. These act geometrically on 7-systolic complexes, i.e. flag simplicial complexes without full cycles of length <7. A leading question is if all hyperbolic groups are residually finite, which was verified by Agol and Wise for cocompactly cubulated groups. But are 7-systolic groups residually finite? Which 3-manifold groups, Artin groups, Coxeter groups and random groups are cocompactly cubulated, or linear?II. A graph is dismantlable if one can reduce it to a single vertex by repeatedly removing vertices that are dominated. Examples are 1-skeleta of CAT(0) cube complexes with added cube diagonals, or weakly-systolic complexes (Chepoi and Osajda). In geometric topology: Kakimizu graf of a knot in the 3-sphere (my work with Schultens), the arc graph, the disc graph and the sphere graph (my work with Hensel and Osajda). I want to exploit this to understand the surface automorphism group, handlebody group, Out(F_n), groups of automorphisms or right-angled Artin groups, hyperbolicity and contractibility of the complexes on which these groups act and realisation theorems.Objectives1.Manifolds and cube complexesContinuing my work with Wise, I would like to characterize 3-manifolds that are cubulated cocompactly. To understand if all 3-manifold groups are linear, for which it remains to study graph manifolds that do not admit a npc metric. With my student Jankiewicz I would like to revisit Wise's cubical small cancellation. Finally, we would like to show that Artin groups of large type and more than 2 generators are not cubulated. 2.Random groupsI would like to find a threshold density for a random group in Gromov density model to be cubulated. With Mackay we want to show that for d<1/4 a random group is cubulated cocompactly. I suspect that for d>1/4 a random group has property (T). Is it then residually finite?3.Coxeter groupsAre Coxeter groups cubulated cocompactly? The second question I am pursuing in Coxeter groups is my work with Caprace to prove Muehlherr's conjecture classifying isomorphisms between Coxeter groups.4.Curve complexDismantlability approach allowed me with Hensel and Webb to give a short proof of the famous Masur-Minksy theorem that the curve graph is hyperbolic. We intend to generalize this to disc graphs and sphere graphs. Is the graph obtained from the curve graph by adding edges between curves intersecting once dismantlable? What is its dimension?5.Dismantlability and realizationWe used dismantlability with Hensel and Osajda to find a combinatorial solution to the Nielsen Realization Problem for surfaces with boundary and to realization problems for the handlebody group and Out(F_n). I would like to obtain a similar result for finite subgroups G of Out(F), where F is any right-angled Artin group. Is there a fixed point of the action of G on some naturally arising Outer space, say the one defined by Crisp, Charney and Vogtmann? I would like to use dismantlability to solve the Nielsen Realization Problem for closed surfaces as well. The strategy is based on verifying that every isometrically embedded subgraph of the 1-skeleton of a CAT(0) cube complex, after adding cube diagonals, is dismantlable.

该方案包括研究几何拓扑和群论问题，如双曲或不动点定理使用组合方法与图论性质称为可拆解性的突出作用。长期目标是：1。找到更多类的双曲群是剩余有限的。在几何拓扑中应用可拆解性。双曲群的例子有几何作用于双曲空间的群、小抵消群、随机群和Haglund和Januszkiewicz-Swiatkowski构造的高维7收缩群。它们在7-收缩复合物上起几何作用，即长度小于7的完整周期的标志简单复合物。一个主要的问题是是否所有双曲群都是剩余有限的，这是由Agol和Wise对紧计算群所证实的。但是7个收缩期群是有限的吗？哪些3-流形群、Artin群、Coxeter群和随机群是紧计算的，还是线性的？2。一个图是可拆解的，如果一个人可以通过反复移除被支配的顶点来将它减少到一个顶点。例如CAT(0)立方体复合物的1-骨架加上立方体对角线，或弱收缩复合物（Chepoi和Osajda）。在几何拓扑中：三球（我与Schultens合作的作品）、弧图、盘图和球图（我与Hensel和Osajda合作的作品）中的一个结的Kakimizu接枝。我想利用这一点来理解曲面自同构群，柄体群，Out(F_n)，自同构群或直角Artin群，这些群所作用的复合体的双曲性和可收缩性，并实现定理。流形和立方复形继续我与Wise的工作，我想描述紧计算的3-流形。为了理解是否所有的3流形群都是线性的，它仍然需要研究不允许npc度量的图流形。和我的学生Jankiewicz一起，我想重温Wise的立方体小消去。最后，我们要证明，大类型和2个以上生成器的Artin群是不被计算的。2.随机群体我想在Gromov密度模型中找到一个随机群体的阈值密度。我们用Mackay证明，当d<1/4时，随机群是紧计算的。我怀疑对于d bb0 1/4，随机群具有性质(T)那么它是剩余有限的吗？考克斯特组考克斯特组是紧密计算的吗？我在Coxeter群中追求的第二个问题是我与Caprace一起证明Muehlherr关于Coxeter群之间同构分类的猜想。曲线复数可拆解性方法让我和Hensel和Webb对著名的Masur-Minksy定理（曲线图是双曲的）给出了一个简短的证明。我们打算把这推广到圆盘图和球图。在一次相交的曲线之间加边得到的曲线图是否可拆解？它的维数是多少？可拆解性和可实现性我们利用Hensel和Osajda的可拆解性，找到了具有边界曲面的Nielsen实现问题的组合解，以及柄体群和Out（F_n）的实现问题的组合解。我想对Out(F)的有限子群G得到一个类似的结果，其中F是任意直角Artin群。在一些自然产生的外空间，比如由Crisp， Charney和Vogtmann定义的外空间，G的作用是否有一个固定点？我也想用可拆解性来解决封闭表面的尼尔森实现问题。该策略基于验证CAT(0)立方体复合体的1-骨架的每个等距嵌入子图，在添加立方体对角线后，是可拆卸的。