Matching Theorems for Systems of a Finitely Generated Coxeter Group

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [3] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. For a recent survey, see Mühlherr [10]. The isomorphism problem for finitely generated Coxeter groups is equivalent to the problem of determining all the automorphism equivalence classes of sets of Coxeter generators for an arbitrary finitely generated Coxeter group. In this paper, we prove a series of matching theorems for two sets of Coxeter generators of a finitely generated Coxeter group that identify common features of the two sets of generators. As an application, we describe an algorithm for finding a set of Coxeter generators of maximum rank for a finitely generated Coxeter group. In §2, we state some preliminary results. In §3, we prove a matching theorem for two systems of a finite Coxeter group. In §4, we prove our Basic Matching Theorem between the sets of maximal noncyclic irreducible spherical subgroups of two systems of a finitely generated Coxeter group. In §5, we study nonisomorphic basic matching. In §6, we prove a matching theorem between the sets of noncyclic irreducible spherical subgroups of two systems of a finitely generated Coxeter group. As an application, we prove the Edge Matching Theorem. In §7, we discuss twisting and flattening visual graph of groups decompositions of Coxeter systems. In §8, we prove the