Quotient Isomorphism Invariants 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 [4] 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. In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W . Each of these invariants is the isomorphism type of a quotient group W/N of W by a characteristic subgroup N . The virtue of these invariants is that W/N is also a Coxeter group. For some of these invariants, the isomorphism problem of W/N is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups. We emphasize that even if the isomorphism problem for finitely generated Coxeter groups is eventually solved, several of the algorithms described in our paper will still be useful because they are computational fast and would most likely be incorporated into an efficient computer program that determines if two finite rank Coxeter systems have isomorphic groups. In §2, we establish notation. In §3, we describe two elementary quotienting operations on a Coxeter system that yields another Coxeter system. In §4, we describe the binary isomorphism invariant of a finitely generated Coxeter group. In §5, we review some matching theorems. In §6, we describe the even isomorphism invariant of a finitely generated Coxeter group. In §7, we define basic characteristic subgroups of a finitely generated Coxeter