Hyperbolicity and dismantlability

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。* *我。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。它们以几何方式作用于7-收缩配合物，即没有长度为1/4的完整循环的标志简单配合物，随机组具有性质(T)。那么它是剩余有限的吗？考克斯特组**考克斯特组是否计算紧凑？我在Coxeter群中追求的第二个问题是我与Caprace一起证明Muehlherr在Coxeter群之间分类同构的猜想。曲线复合体**可拆解性方法让我与Hensel和Webb一起给出了著名的Masur-Minksy定理的简短证明，即曲线图是双曲的。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？可拆解性与实现**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。

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 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.