Classify hyperbolic groups with all infinite-index subgroups free (Gromov–Whyte)

Determine whether an infinite hyperbolic group G in which every subgroup of infinite index is free must be either a free group or a surface group.

Background

The paper answers this classification affirmatively for cubulated hyperbolic groups, producing one-ended quasiconvex subgroups unless the group is free or a surface group. Extending this classification to all hyperbolic groups remains open.

This question reiterates problems posed by Gromov and Whyte (Questions 1.7 and 1.11 in Bestvina’s list), and resolving it would have major implications, including connections to the existence of surface subgroups in one-ended hyperbolic groups.