Hyperbolicity and dismantlability

该计划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，其中突出的作用是称为可拆卸性的图论属性。长期目标是： I. 找到更多类残差有限的双曲群。 二.在几何拓扑中应用可拆卸性。 I. 双曲群的例子有几何作用于双曲空间的群、小抵消群、随机群以及由 Haglund 和 Januszkiewicz-Swiatkowski 构造的高维 7 收缩群。这些在几何上作用于 7 收缩复合体，即标记没有长度 <7 的完整周期的单纯复合体。一个主要问题是是否所有双曲群都是残差有限的，Agol 和 Wise 已针对协紧立方群验证了这一点。 但是 7 收缩群是剩余有限的吗？哪些 3 流形群、Artin 群、Coxeter 群和随机群是紧立方的，还是线性的？ 二.如果可以通过重复删除受支配的顶点来将图缩减为单个顶点，则图是可拆卸的。例如，具有添加立方体对角线的 CAT(0) 立方体复合体的 1 骨架，或弱收缩复合体（Chepoi 和 Osajda）。在几何拓扑中：3-球体中结的 Kakimizu 绘图（我与 Schultens 合作）、弧形图、圆盘图和球体图（我与 Hensel 和 Osajda 合作）。我想利用它来理解表面自同构群、手柄群、Out(F_n)、自同构群或直角 Artin 群、这些群作用的复合体的双曲性和可收缩性以及实现定理。 目标 1.流形和立方体复合体 继续我与 Wise 的合作，我想描述 3 流形的共同紧致立方体的特征。要了解是否所有 3 流形群都是线性的，为此需要研究不接受 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，我们想证明大型类型和超过 2 个生成器的 Artin 群不会被累积。 2.随机分组 我想找到要计算的格罗莫夫密度模型中随机组的阈值密度。通过 Mackay，我们想要证明，对于 d<1/4，随机组是协紧累积的。我怀疑对于 d>1/4，随机组具有属性 (T)。那么它是剩余有限的吗？ 3.考克塞特群 Coxeter 群是紧立方的吗？我在 Coxeter 群中追求的第二个问题是我与 Caprace 一起证明 Muehlherr 对 Coxeter 群之间同构进行分类的猜想。 4.曲线复合体 可拆卸性方法使我能够与亨塞尔和韦伯一起给出著名的马苏尔-明克西定理的简短证明，即曲线图是双曲的。我们打算将其推广到圆盘图和球体图。由曲线图通过在相交的曲线之间添加边得到的图形是否可拆卸？它的维度是多少？ 5.可拆解与实现 我们使用 Hensel 和 Osajda 的可拆卸性来找到具有边界的曲面的 Nielsen 实现问题以及手柄组和 Out(F_n) 的实现问题的组合解决方案。我想为 Out(F) 的有限子群 G 获得类似的结果，其中 F 是任何直角 Artin 群。 G 在某些自然产生的外层空间上的作用是否存在一个固定点，比如 Crisp、Charney 和 Vogtmann 定义的那个？ 我也想使用可拆卸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 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。

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.