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We prove that certain families of Coxeter groups and inclusions $W_1hookrightarrow W_2hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the basic construction, and then use Daviss description of the basic construction as an increasing union of chambers to deduce the required connectivity.
We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the $K(pi,1)$ conjecture holds for the associated family of Artin groups this establishes homological stability for these
We show that the Iwahori-Hecke algebras H_n of type A_{n-1} satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaokas homological stability result for the symmetric groups in the
We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is enti
In this paper we study homological stability for spaces ${rm Hom}(mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $ngeqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any
Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one use Coxeter groups to construct interesting examples of discrete subgroups of Lie group.