Surface subgroups from linear programming

Danny Calegari, Alden Walker
2015 Duke mathematical journal  
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 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
more » ... 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.
doi:10.1215/00127094-2877511 fatcat:qim6ewgemjcy3dgr5yo5expixe