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Surface subgroups from linear programming. (English) Zbl 1367.20026

Summary: 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.

MSC:

20E05 Free nonabelian groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
57M07 Topological methods in group theory
20P05 Probabilistic methods in group theory
90C05 Linear programming
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