Strong accessibility for finitely presented groups

TL;DR

This paper introduces a hierarchy framework for finitely presented groups, proving finiteness under certain conditions, with applications to hyperbolic groups and subgroups of SL(n,Z).

Contribution

It defines a new (relative) slender JSJ hierarchy for finitely presented groups and proves its finiteness under specific algebraic conditions.

Findings

Finiteness of the hierarchy for certain groups

Applicability to hyperbolic groups without 2-torsion

Relevance to subgroups of SL(n,Z)

Abstract

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.