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 SLn(Z) are both finite.

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinct pair of conjugacy classes.

Abstract. We prove Wise's W-cycles conjecture: Consider a compact graph Γ ′ immersing into another graph Γ. For any immersed cycle Λ ∶ S → Γ, we consider the map Λ ′ from the circular components S of the pullback to Γ ′ . Unless Λ ′ is reducible, the degree of the covering map S → S is bounded above by minus the Euler characteristic of Γ ′ . As a corollary, any nitely generated subgroup of a one-relator group has nitely generated Schur multiplier.

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be a free group, and let w ∈ F be a non-primitive element. The primitivity rank of w, π(w), is the smallest rank of a subgroup of F containing w as an imprimitive element. Then any subgroup of the onerelator group G = F/ w generated by fewer than π(w) elements is free. In particular, if π(w) > 2 then G doesn't contain any Baumslag-Solitar groups.The hypothesis that π(w) > 2 implies that the presentation complex X of the one-relator group G has negative immersions: if a compact, connected complex Y immerses into X and χ(Y ) ≥ 0 then Y is Nielsen equivalent to a graph.The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus' Freiheitssatz and theorems of Lyndon, Baumslag, Stallings and Duncan-Howie.The dependence theorem strengthens Wise's w-cycles conjecture, proved independently by the authors and Helfer-Wise, which implies that the one-relator complex X has non-positive immersions when π(w) > 1.

This is the third paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. We give generalizations of the well known fact that a nontrivial commutator in a free group is not a proper power to both graphs of free groups over cyclic subgroups and freely decomposable groups. None of the paper is specifically about limit groups.