On the Isomorphism Problem for Finitely Generated Coxeter Groups. I Basic Matching

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. In this paper, we determine some strong necessary conditions for two Coxeter groups to be isomorphic in terms of their subgroups and quotient groups. In Part I of our paper, we prove a matching theorem for maximal rank irreducible noncyclic spherical subgroups of isomorphic Coxeter groups. In Part II of our paper, we describe an algorithm for finding a presentation graph of maximum rank for a finitely generated Coxeter group and prove that the maximum rank presentation graphs of isomorphic finitely generated Coxeter groups all have the same number of vertices and the same number of k-labeled edges for each integer k ≥ 2. In §2, we state some preliminary results. In §3, we prove a matching theorem for systems of a finite Coxeter group. In §4, we prove our Basic Matching Theorem, Theorem 4.19. In §5, we study nonisomorphic basic matching. In §6, we prove a matching theorem for irreducible noncyclic spherical subgroups of isomorphic Coxeter groups. Part I ends with the Edge Matching Theorem for isomorphic Coxeter groups.