# Groups, Geometry, and Dynamics

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**Volume 3, Issue 1, 2009, pp. 173–198**

**DOI: 10.4171/GGD/53**

Published online: 2009-03-31

Visual decompositions of Coxeter groups

Michael Mihalik^{[1]}and Steven Tschantz

^{[2]}(1) Vanderbilt University, Nashville, United States

(2) Vanderbilt University, Nashville, United States

A Coxeter system is an ordered pair (`W`,`S`) where `S` is the
generating set in a particular type of presentation for the
Coxeter group `W`. A subgroup of `W` is called special if it is
generated by a subset of `S`. Amalgamated product decompositions
of a Coxeter group having special factors and special amalgamated
subgroup are easily recognized from the presentation of the
Coxeter group. If a Coxeter group is a subgroup of the
fundamental group of a given graph of groups, then the Coxeter
group is also the fundamental group of a graph of special
subgroups, where each vertex and edge group is a subgroup of a
conjugate of a vertex or edge group of the given graph of groups.
A vertex group of an arbitrary graph of groups decomposition of a
Coxeter group is shown to split into parts conjugate to special
groups and parts that are subgroups of edge groups of the given
decomposition. Several applications of the main theorem are
produced, including the classification of maximal FA subgroups of
a finitely generated Coxeter group as all conjugates of certain
special subgroups.

*Keywords: *Coxeter group, group actions on trees, graph of group decompositions, FA groups, accessibility of groups

Mihalik Michael, Tschantz Steven: Visual decompositions of Coxeter groups. *Groups Geom. Dyn.* 3 (2009), 173-198. doi: 10.4171/GGD/53