论文标题
连接图和Coxeter组的拓扑边界
Topological boundaries of connected graphs and Coxeter groups
论文作者
论文摘要
我们介绍并研究与连接的根图相关的某些拓扑空间。这些空间反映了基础图的组合和秩序理论特性，并在双曲线图与格罗莫夫双曲线紧凑型的情况下相关。对于有限级Coxeter组的Cayley图，它们尤其可以处理。在这种情况下，我们谈到了Coxeter组的紧凑和边界。事实证明，Coxeter群在其Cayley图上的规范作用引起了对紧凑型和边界的自然作用。由此我们推断出，在这种情况下，我们的构建与定义的Caprace和Lécureux的空间一致。我们进一步证明了该动作的不适当性，我们表征了何时紧凑型在无穷大小，并且我们研究了Coxeter组的类别，从而在Furstenberg的意义上是该作用是拓扑边界作用。 本文的第二部分介绍了我们的结果在（Iwahori）Hecke代数的研究中的应用。这些是Coxeter组的组代数的某些变形。我们首先研究了Hecke C $^\ AST $ -Algebras的嵌入，并证明了某些类别的Hecke-von Neumann代数为Akemann-Ostrand的财产。最后，我们利用与Kalantar-Kennedy对C $^\ ast $ - 舒张性问题的方法广泛相关的结果，以研究与Hecke代数相关的操作员代数的简单性和注入式信封。
We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. In that context we speak of the compactification and the boundary of the Coxeter group. As it turns out, the canonical action of the Coxeter group on its Cayley graph induces a natural action on the compactification and the boundary. From this we deduce that in this case our construction coincides with spaces defined Caprace and Lécureux. We further prove the amenability of the action, we characterize when the compactification is small at infinity and we study classes of Coxeter groups for which the action is a topological boundary action in the sense of Furstenberg. The second part of the paper deals with the applications of our results to the study of (Iwahori) Hecke algebras. These are certain deformations of group algebras of Coxeter groups. We first study embeddings of Hecke C$^\ast$-algebras and prove property Akemann-Ostrand for a certain class of Hecke-von Neumann algebras. Lastly, we make use of results that are widely related to Kalantar-Kennedy's approach to the C$^\ast$-simplicity problem to study the simplicity and injective envelopes of operator algebras associated with Hecke algebras.
