## Algebraic & Geometric Topology

### Matching theorems for systems of a finitely generated Coxeter group

#### Abstract

We study the relationship between two sets $S$ and $S′$ of Coxeter generators of a finitely generated Coxeter group $W$ by proving a series of theorems that identify common features of $S$ and $S′$. 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 $S′$ have maximum rank, then there is a bijection between the complete subsets of $S$ and the complete subsets of $S′$ so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of $(W,S)$ and $(W,S′)$ have the same multiset of entries.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 2 (2007), 919-956.

Dates
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796710

Digital Object Identifier
doi:10.2140/agt.2007.7.919

Mathematical Reviews number (MathSciNet)
MR2336245

Zentralblatt MATH identifier
1134.20045

Keywords
Coxeter groups

#### Citation

Mihalik, Michael; Ratcliffe, John G; Tschantz, Steven T. Matching theorems for systems of a finitely generated Coxeter group. Algebr. Geom. Topol. 7 (2007), no. 2, 919--956. doi:10.2140/agt.2007.7.919. https://projecteuclid.org/euclid.agt/1513796710

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