Hyperbolicity and dismantlability

该课程包括使用组合方法研究几何拓扑和群论问题，如双曲性或不动点定理，其中突出的作用是一种称为可拆卸性的图论性质。我们的长远目标是： I.找到更多剩余有限的双曲群。 二、将可拆卸性应用于几何拓扑学。 I.双曲群的例子是几何作用在双曲空间上的群、小消去群、随机群和由Haglund和Januszkiewicz-Swiatkowski构造的高维7-收缩群。它们几何地作用于7-收缩复形，即没有长度为&lt；7的全圈的旗形单纯复形。一个主要问题是，是否所有双曲群都是剩余有限的，这一点已被Agol和Wise对余紧立方群所证实。但是7-收缩期群是剩余有限的吗？哪个3-流形群，Artin群，Coxeter群和随机群是上紧的，还是线性的？ Ii.如果一个图可以通过重复删除被支配的顶点而减少到单个顶点，则该图是可拆卸的。例如CAT(0)立方体复合体的1-骨架加上立方对角线，或弱收缩复合体(Chepoi和Osajda)。几何拓扑学：三维球面上纽结的Kakimizu Graf(我与Schulten的工作)、弧图、圆盘图和球面图(我与Hensel和Osajda的工作)。我想利用这一点来理解曲面自同构群、HandleBody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缩性以及实现定理。 目标 1.流形和立方体复形 继续我与Wise的工作，我想刻画3-流形是上紧的。要了解是否所有的3-流形群都是线性的，还需要研究不允许NPC度量的图流形。和我的学生Jankiewicz一起，我想重温怀斯的立方体小取消。最后，我们要证明的是，大型群和超过2个生成元的Artin群是不分块的。 2.随机分组 我想为格罗莫夫密度模型中的一个随机群找到一个阈值密度。有了Mackay，我们想要证明：对于d&lt；1/4，随机群是上紧的。我猜想对于d&gt；1/4，一个随机群具有性质(T)。那么它是剩余有限的吗？ 3.Coxeter组 Coxeter组是不是密实的？我在Coxeter群中追求的第二个问题是我和Caprace一起证明了Muehlherr猜想，该猜想将Coxeter群之间的同构分类。 4.曲线复合体 可分解性方法允许我与Hensel和Webb一起给出著名的Masur-Minksy定理的简短证明，即曲线图是双曲的。我们打算将其推广到圆盘图和球形图。通过在相交的曲线之间添加边而获得的曲线图是否可以一次拆卸？它的尺寸是多少？ 5.可分解性与实现 利用Hensel和Osajda的可分解性，求出了带边界曲面的Nielsen实现问题的组合解，以及手体组和OUT(F_N)的实现问题。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。在克里斯普、查尼和沃格特曼定义的某个自然形成的外层空间上，G的作用是否存在一个固定点？ 我也想用可拆卸性来解决闭合曲面的尼尔森实现问题。该策略的基础是验证在添加立方体对角线后，CAT(0)立方体复合体的1-骨架的每个等距嵌入子图都是可拆卸的。

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.