Surface subgroups from linear programming

Abstract

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ 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 $2$ free group sending $a$ to $ab$ and $b$ to $ba$; 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 $2$) 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.