A finitely generated group has a natural geometry: the word metric on its Cayley graph. A one-ended hyperbolic group is, coarsely, a connected space with no large-scale bottlenecks that disconnect it. But some of these spaces can still be separated by removing a relatively small subset — subexponential in size compared to the balls it divides.
Bensaid, Genevois, and Tessera (arXiv:2603.17852) characterize exactly when this happens. A one-ended hyperbolic group that is not virtually a surface group can be coarsely separated by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. The geometric property (coarse separation) is equivalent to the algebraic property (splitting).
The equivalence is sharp. If the group doesn't split over a virtually cyclic subgroup, then any subset that separates large balls must itself have exponential size — the cut is as expensive as the space it divides. The proof establishes polynomial lower bounds on separation profiles for non-splitting groups, quantifying how the minimum cut-set size grows with the radius of the balls being separated.
Surface groups are the exception. Their two-dimensional nature gives them separation properties that don't reduce to algebraic splittings, which is why they must be excluded from the characterization.
The result connects two scales of understanding: the fine algebraic structure (virtual splittings over small subgroups) and the coarse geometric structure (how efficiently the space can be disconnected). They see the same thing. The cheapest way to cut a hyperbolic group in half is determined by its algebraic anatomy, and the price is either subexponential or exponential with nothing in between.