Algebraic & Geometric Topology

Matching theorems for systems of a finitely generated Coxeter group

Michael Mihalik, John G Ratcliffe, and Steven T Tschantz

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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

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

Received: 31 March 2006
First available in Project Euclid: 20 December 2017

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Coxeter groups


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.

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