Duke Mathematical Journal

Surface subgroups from linear programming

Danny Calegari and Alden Walker

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We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive b2 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.

Article information

Duke Math. J., Volume 164, Number 5 (2015), 933-972.

First available in Project Euclid: 7 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 57M07: Topological methods in group theory 20P05: Probabilistic methods in group theory [See also 60Bxx]

surface subgroup free group endomorphism hyperbolic group Gromov’s question


Calegari, Danny; Walker, Alden. Surface subgroups from linear programming. Duke Math. J. 164 (2015), no. 5, 933--972. doi:10.1215/00127094-2877511. https://projecteuclid.org/euclid.dmj/1428419276

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