A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost) finitely presented groups and show that it is finite, provided the group in question doesn't contain any slender subgroups with infinite dihedral quotients and satisfies an ascending chain condition on certain chains of subgroups of edge groups. As a corollary, slender JSJ hierarchies of hyperbolic groups which are (virtually) without $2$--torsion and finitely presented subgroups of SL(n,Z) are both finite.
- Pub Date:
- February 2013
- Mathematics - Group Theory
- 29 pages, 11 figures. author note: the arXiv admin note refers to a quotation. arXiv admin note: text overlap with arXiv:math/0507424 by other authors