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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611626
• 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. Objectives 1.Manifolds and cube complexes Continuing 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 groups I 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 groups Are 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 complex Dismantlability 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 realization We 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.

该计划包括研究几何拓扑和群论问题，如双曲性或不动点定理，使用组合方法，具有称为不稳定性的图论性质的突出作用。长期目标是： I.找到更多剩余有限的双曲群类。 二.在几何拓扑中应用不稳定性。 I.双曲群的例子有几何上作用于双曲空间的群、小消去群、随机群和Haglund和Januszkiewicz-Swiatkowski构造的高维7-收缩群。它们在几何上作用于7-收缩复形，即没有长度<7的完整循环的标志单纯复形。一个主要的问题是，如果所有的双曲群是剩余有限的，这是由Agol和Wise验证的cocompensarcubulated群体。但是7收缩组是剩余有限的吗？哪些3-流形群、Artin群、Coxeter群和随机群是余紧群或线性群？ 二.一个图是可分解的，如果一个人可以减少它的一个单一的顶点，通过反复删除顶点的支配。例子是CAT（0）立方体复合物的1-β，加上立方体对角线，或弱收缩复合物（Chepoi和Osajda）。在几何拓扑学中：Kakimizu图的一个结在3球（我的工作与Schultens），弧图，盘图和球图（我的工作与Hensel和Osajda）。我想利用这一点来理解曲面自同构群、三体群、Out（F_n）、自同构群或直角Artin群、这些群作用于其上的复形的双曲性和可收缩性以及实现定理。 目标 1.流形和立方复形 继续我的工作与明智的，我想刻画3-流形是cubulated cocompensation。为了理解是否所有的3-流形群都是线性的，这就需要研究不允许npc度量的图流形。我想和我的学生Jankiewicz一起重温怀斯的立方体小取消。最后，我们想证明大型的Artin群和多于2个生成元的Artin群是不可求的。 2.随机分组 我想找到一个阈值密度的随机组在格罗莫夫密度模型进行cubulated。与Mackay一起，我们想证明，对于d<1/4，一个随机群是cubulated余紧群。我猜想，对于d>1/4，一个随机群具有性质（T）。那么它是剩余有限的吗？ 3. Coxeter群 Coxeter群是共生的吗？第二个问题，我追求在考克斯特群体是我的工作与Caprace证明Muehlherr猜想分类之间的同构考克斯特群体。 4.曲线复形 可分解性方法使我与亨泽尔和韦伯给一个简短的证明著名的马苏尔-明克西定理，曲线图是双曲的。我们打算将其推广到圆盘图和球图。通过在相交一次的曲线之间添加边而从曲线图获得的图是可验证的吗？它的尺寸是多少？ 5.可拆卸性和实现 我们利用Hensel和Osajda的可解性找到了带边界曲面的Nielsen实现问题和三体群及Out（F_n）的实现问题的组合解。我想对Out（F）的有限子群G得到类似的结果，其中F是任意直角Artin群。G在自然产生的外层空间上的作用是否有一个不动点，比如说克里斯普、查尼和沃格特曼所定义的不动点？ 我也想用可解性来解决封闭曲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之后，是可分解的。