Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

TL;DR

This paper demonstrates that cyclic subgroup separability is maintained under exponential completion in certain classes of groups, including coherent RAAGs, leading to the conclusion that limit groups over these RAAGs are cyclic subgroup separable.

Contribution

It establishes the preservation of cyclic subgroup separability under exponential completion for groups including coherent RAAGs, answering a previously open question.

Findings

Cyclic subgroup separability is preserved under exponential completion.

Limit groups over coherent RAAGs are cyclic subgroup separable.

The word problem is solvable for these groups.

Abstract

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.