We study the relationship between two sets and of Coxeter generators of a finitely generated Coxeter group by proving a series of theorems that identify common features of and . We describe an algorithm for constructing from any set of Coxeter generators of a set of Coxeter generators of maximum rank for .
A subset of is called complete if any two elements of generate a finite group. We prove that if and have maximum rank, then there is a bijection between the complete subsets of and the complete subsets of so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of and have the same multiset of entries.
"Matching theorems for systems of a finitely generated Coxeter group." Algebr. Geom. Topol. 7 (2) 919 - 956, 2007. https://doi.org/10.2140/agt.2007.7.919