Visual JSJ Decompositions of Coxeter Groups
by
Michael L. Mihalik
Vanderbilt University
Coauthors: Steven Tschantz

A Coxeter group is a finitely presented group with Coxeter presentation

<s1, ... , sn: si2=1 for alli, (sisj)mij=1>

where m_ij is an integer >= 2 for some set of pairs i < j. The pair (G, S) is a Coxeter System. If (G, S) is a Coxeter system and A subset S then <A> is called a visual subgroup of (G, S). A JSJ decomposition of a group G is a graph of groups decomposition of G such that each edge group is virtually cyclic and each vertex group is indecomposable in the sense that it does not split along a virtually cyclic group in a way that is compatible with the given decomposition. This talk will center around JSJ decompositions and how visual they must be.

Theorem. Suppose G is a 1-ended finitely generated Coxeter group having graph of groups decomposition \Lambda where each edge group is 2-ended. Then G has a visual JSJ decomposition \Psi where each vertex group of \Psi is a subgroup of a conjugate of a vertex group of \Lambda.

In some sense this says that any partial construction of a JSJ decomposition for a Coxeter group is refined by a visual JSJ decomposition.

Date received: September 29, 2000