Surface subgroups from linear programming

TL;DR

This paper demonstrates the existence of surface subgroups in certain graphs of free groups using combinatorial, geometric, and linear programming methods, and characterizes the Gromov norm unit ball as a rational polyhedron.

Contribution

It introduces new techniques to identify surface subgroups in complex free group constructions and characterizes the Gromov norm unit ball in these contexts.

Findings

Certain graphs of free groups contain surface subgroups.

The Gromov norm unit ball is a finite-sided rational polyhedron.

Every rational class is virtually represented by an extremal surface subgroup.

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 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.