arXiv subjects

William D. Cohen

Publications and source records attributed to William D. Cohen.

4 recordsLinked to original sources

arXiv2026

Quasi-convex Splittings of Acylindrical Graphs of Locally Finite-Height Groups

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.

math.GR↗

arXiv2026

Groups Acting Acylindrically on Trees

We develop a notion of groups that act acylindrically and non-elementarily on simplicial trees, which we call acylindrically arboreal groups. We then prove a complete classification of when graph products of groups and the fundamental groups of certain hyperbolic $3$-manifolds are acylindrically arboreal, and use these classifications to provide examples of acylindrically hyperbolic groups that have actions on trees but have no non-elementary acylindrical actions on trees.

math.GR↗

arXiv2025

Acylindrical Visual Splittings and The Tits Alternative For Artin Groups

We give a necessary and sufficient condition on a visual splitting of an Artin group satisfying the conditions of two well known conjectures to be acylindrical, and demonstrate how this can be used to provide a large class of novel examples of Artin groups that satisfy the Tits alternative.

math.GR↗

arXiv2024

Cohomological Separability of Baumslag--Solitar groups and Their Generalisations

A group $Γ$ has separable cohomology if the profinite completion map $ι\colon Γ\to \widehatΓ$ induces an isomorphism on cohomology with finite coefficient modules. In this article, cohomological separability is decided within the class of generalised Baumslag--Solitar groups, i.e. graphs of groups with infinite cyclic fibers. Equivalent conditions are given both explicitly in terms of the defining graph of groups and in terms of the induced topology on vertex groups. Restricted to the class of Baumslag--Solitar groups, we obtain a trichotomy of cohomological separability and cohomological dimension of the profinite completions. In particular, this yields examples of non-residually-finite one-relator groups which have separable cohomology, and examples which do not.

math.GR↗