Local quasiconvexity in graphs of free groups with cyclic edge groups

Question

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?

Answer 1

You're right that hyperbolic graphs of free groups with cyclic edge groups are locally quasiconvex. This can be proved by combining subgroup separability with results about combination of quasiconvex subgroups, such as the one proved in the relatively hyperbolic context by Eduardo Martinez-Pedroza.

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).

You also write 'It seems to me that any f.p. subgroup of these groups has to be quasiconvex.' I'm not sure what groups you mean by 'these groups', but perhaps it's worth pointing out that Noel Brady constructed a hyperbolic group (with a quasiconvex hierarchy) with a finitely presented, non-hyperbolic (in particular, non-quasiconvex) subgroup.