#### Volume 7, issue 2 (2007)

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Matching theorems for systems of a finitely generated Coxeter group

### Michael L Mihalik, John G Ratcliffe and Steven T Tschantz

Algebraic & Geometric Topology 7 (2007) 919–956
 arXiv: math.GR/0501075
##### Abstract

We study the relationship between two sets $S$ and ${S}^{\prime }$ of Coxeter generators of a finitely generated Coxeter group $W$ by proving a series of theorems that identify common features of $S$ and ${S}^{\prime }$. We describe an algorithm for constructing from any set of Coxeter generators $S$ of $W$ a set of Coxeter generators $R$ of maximum rank for $W$.

A subset $C$ of $S$ is called complete if any two elements of $C$ generate a finite group. We prove that if $S$ and ${S}^{\prime }$ have maximum rank, then there is a bijection between the complete subsets of $S$ and the complete subsets of ${S}^{\prime }$ so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of $\left(W,S\right)$ and $\left(W,{S}^{\prime }\right)$ have the same multiset of entries.

Coxeter groups