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?