# Local quasiconvexity in graphs of free groups with cyclic edge groups

In Wise1 Wise shows that hyperbolic graphs of free groups with cyclic edge groups are subgroup separable. In Hsu-Wise these are shown to be cubulated, and by Agol they're virtually special, so quasiconvex subgroups by Haglund-Wise are virtual retracts. This motivates the following:

• Are hyperbolic graphs of free groups with cyclic edge groups locally quasiconvex? Edit: Section 4 of Bigdely-Wise offers some evidence in favour of this.

• What about hyperbolic groups with a quasiconvex hierarchy? Edit: No. In Agol it is pointed out in the introduction that by Haglund-Wise hyperbolic special groups have a quasiconvex hierarchy, but on the other hand some two generated $C'(1/6)$ groups which are hyperbolic and cubulated, by Wise2, are free-by-cyclic. So have a distorted free group. There is also Hagen and Wise's cubulation of hyperbolic free-by-Z groups.

It seems to me that any f.p. subgroup of these groups has to be quasiconvex.

• While I'm at it, is there a connection between subgroup separability and coherence?

You ask whether coherence and subgroup separability are related. Heuristically, the answer is 'yes': one often proves coherence by finding a compact core for a covering space, and one proves subgroup separability by somehow completing a compact core to a finite-sheeted covering space. From this point of view, subgroup separability looks stronger than coherence, and there are plenty of examples that demonstrate this; perhaps the simplest is the Baumslag--Solitar group $BS(1,2)$, which is coherent but not subgroup separable.
I believe there probably are subgroup separable groups that aren't coherent -- for instance, if you apply the Rips construction to $\mathbb{Z}$, you obtain an incoherent group with no obvious inseparable subgroups -- but I don't think any examples are known to exist, mainly because there are very few examples of groups that are genuinely subgroup separable (as opposed to just QCERF, say).