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?

• 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.

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?

I looked for references or counterexamples, but couldn't find any. Hagen and Wise's cubulation of hyperbolic free-by-Z groups, however, shows that virtually special groups need not be locally quasiconvex.

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?

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?

• What about hyperbolic groups with a quasiconvex hierarchy?

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?

I looked for references or counterexamples, but couldn't find any. Hagen and Wise's cubulation of hyperbolic free-by-Z groups, however, shows that virtually special groups need not be locally quasiconvex.

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?

• 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.

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?

I looked for references or counterexamples, but couldn't find any. Hagen and Wise's cubulation of hyperbolic free-by-Z groups, however, shows that virtually special groups need not be locally quasiconvex.