JSJ-Decompositions of Coxeter Groups over Virtually Abelian Splittings

The idea of “JSJ-decompositions” for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of 2-sided, incompressible tori that separate a closed irreducible 3-manifold into pieces with strong topological structure. Sela introduced the idea of JSJ-decompositions for groups, an idea that has flourished in a variety of directions. The general idea is to consider a certain class G of groups and splittings of groups in G by groups in another class C. E.g. Rips and Sela considered splittings of finitely presented groups by infinite cyclic groups. For an arbitrary group G in G, the goal is produce a unique reduced graph of groups decomposition Ψ of G with edge groups in C so that Ψ reveals all reduced graph of groups decompositions of G with edge groups in C. More specifically, if H is a vertex group of Ψ then either there is no C-group that splits both G and H, or H has a special “surface grouplike” structure. Vertex groups of the second type are standardly called