Atlas: Visual JSJ Decompositions of Coxeter Groups by Michael L. Mihalik

Atlas home || Conferences | Abstracts | about Atlas

G^3, Special Session in Geometric Group Theory
January 10-13, 2001
part of the AMS/MAA joint meeting
New Orleans, LA, USA

Organizers
Phil Bowers, Martin Bridson, Stephen Brick, Jon Corson, Igor Mineyev

View Abstracts
Conference Homepage

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, ... , s_n: s_i²=1 for alli, (s_is_j)^m_ij=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