Duke Mathematical Journal

Surface subgroups from linear programming

Danny Calegari and Alden Walker

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Abstract

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

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

Dates
First available in Project Euclid: 7 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1428419276

Digital Object Identifier
doi:10.1215/00127094-2877511

Mathematical Reviews number (MathSciNet)
MR3332895

Zentralblatt MATH identifier
1367.20026

Subjects
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]

Keywords
surface subgroup free group endomorphism hyperbolic group Gromov’s question

Citation

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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