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。长期目标是：**i.找到更多剩余有限的双曲群。*ii.在几何拓扑学中应用可分解性。**I.双曲群的例子有几何作用在双曲空间上的群、小消去群、随机群以及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1/4长满圈的旗形单纯复形。随机群具有性质(T)。那么它是剩余有限的吗？**3.Coxeter群**Coxeter群是上紧的吗？我在Coxeter群中追求的第二个问题是我与Caprace一起证明了Muehlherr将Coxeter群之间的同构分类的猜想。**4曲线复数**可分解性方法使我能够与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？它的维度是多少？**5.可分解性和实现**我们利用Hensel和Osajda的可分解性，求出了有边界曲面的Nielsen实现问题的组合解，以及手体组和外(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。G在某些自然产生的外层空间上的作用是否有一个固定点，比如Crisp，Charney和Vogtmann定义的那个？*我也想用可拆卸性来解决闭合曲面的Nielsen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 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.