We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive obtained by doubling free groups along collections of subgroups and groups obtained by “random” ascending HNN (Higman–Neumann–Neumann) extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank free group sending to and to ; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension ) of a double of a free group along a collection of subgroups is a finite-sided rational polyhedron and that every rational class is virtually represented by an extremal surface subgroup. These results are obtained by a mixture of combinatorial, geometric, and linear programming techniques.
"Surface subgroups from linear programming." Duke Math. J. 164 (5) 933 - 972, 1 April 2015. https://doi.org/10.1215/00127094-2877511