Quasi-convex Splittings of Acylindrical Graphs of Locally Finite-Height Groups
William D. Cohen
TL;DR
This work addresses whether every non-elementary acylindrical action of a finitely presented group on a tree can be dominated by a quasi-convex or finite-height acylindrical splitting with finitely generated edge stabilisers. It introduces the notion of locally finite-height vertex stabilisers and proves that, under this condition, the Dunwoody--Sageev resolution yields a dominating, non-elementary finite-height acylindrical splitting with finitely generated edge stabilisers; in hyperbolic settings this leads to quasi-convex splittings and virtual compact specialness. The paper also provides important applications to subgroups of one-relator groups with acylindrical Magnus hierarchies, showing they admit quasi-convex hierarchies and are virtually compact special, and it furnishes a counterexample illustrating the limits of preserving acylindricity under the resolution in general. Collectively, these results tie acylindrical actions, finite height, and quasi-convex splittings together, with significant implications for Wise’s program and virtual Haken phenomena in hyperbolic and one-relator groups. An explicit construction demonstrates that the DS-resolution need not preserve acylindricity without the proposed local hypotheses, motivating the precise hypotheses used throughout the paper and inviting further questions about domination in broader contexts.
Abstract
We find a condition on the acylindrical action of a finitely presented group on a simplicial tree which guarantees that this action will be dominated by an acylindrical action with finitely generated edge stabilisers, and find the first example of an action of a finitely presented group where there is no such dominating action. As a consequence, we show that any finitely presented group that admits a decomposition as an acylindrical graph of (possibly infinitely generated) free groups is virtually compact special, and that every finitely generated subgroup of a one-relator group with an acylindrical Magnus hierarchy is virtually compact special.