Duke Mathematical Journal

Surface subgroups from linear programming

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 (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

https://projecteuclid.org/euclid.dmj/1428419276

Digital Object Identifier
doi:10.1215/00127094-2877511

Mathematical Reviews number (MathSciNet)
MR3332895

Zentralblatt MATH identifier
1367.20026

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

References

• [1] C. Bavard, Longueur stable des commutateurs, Enseign. Math. 37 (1991), 109–150.
• [2] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. Math. 135 (1992), 1–51.
• [3] R. Brooks, “Some remarks on bounded cohomology, Riemann surfaces and related topics” in Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981, 53–63.
• [4] D. Calegari, Surface subgroups from homology, Geom. Topol. 12 (2008), 1995–2007.
• [5] D. Calegari, scl, MSJ Memoirs, 20, Math. Soc. Japan, Tokyo, 2009.
• [6] D. Calegari, Stable commutator length is rational in free groups, J. Amer. Math. Soc. 22 (2009), 941–961.
• [7] D. Calegari and A. Walker, Isometric endomorphisms of free groups, N. Y. J. Math. 17 (2011), 713–743.
• [8] D. Calegari, Random rigidity in the free group, Geom. Topol. 17, (2013), 1707–1744.
• [9] D. Calegari, Surface subgroups from linear programming, version 1, preprint, arXiv:1212.2618v2 [math.GR].
• [10] D. Calegari, scallop, computer program, https://github.com/aldenwalker/scallop (accessed 20 January 2015).
• [11] J. Crisp, M. Sageev, and M. Sapir, Surface subgroups of right-angled Artin groups, Internat. J. Algebra Comput. 18 (2008), 443–491.
• [12] M. Culler, Using surfaces to solve equations in free groups, Topology 20 (1981), 133–145.
• [13] GNU Linear Programming Kit, Version 4.45, http://www.gnu.org/software/glpk/glpk.html (accessed 20 January 2015).
• [14] C. Gordon, D. Long, and A. Reid, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004), 135–148.
• [15] C. Gordon and H. Wilton, On surface subgroups of doubles of free groups, J. Lond. Math. Soc. 82 (2010), 17–31.
• [16] R. Grigorchuk, “Some results on bounded cohomology” in Combinatorial and Geometric Group Theory (Edinburgh, 1993), LMS Lecture Note Ser. 204, Cambridge Univ. Press, Cambridge, 1995, 111–163.
• [17] M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. no. 56 (1982), 5–99.
• [18] Gurobi Optimization, Inc., Gurobi Optimizer Reference Manual (2012), available at http://www.gurobi.com (accessed 27 January 2015).
• [19] J. Kahn and V. Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. (2) 175 (2012), 1127–1190.
• [20] S.-H. Kim and S.-I. Oum, Hyperbolic surface subgroups of one-ended doubles of free groups, J. Topology 7 (2014), 927–947.
• [21] S.-H. Kim and H. Wilton, Polygonal words in free groups, Q. J. Math. 63 (2012), 399–421.
• [22] R. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Diff. Geom. 27 (1988), 35–53.
• [23] P. Reynolds, Dynamics of Irreducible Endomorphisms of $F_{n}$, preprint, arXiv:1008.3659v4 [math.GR].
• [24] A. Rhemtulla, A problem of bounded expressibility in free products, Proc. Cambridge Philos. Soc. 64 (1968), 573–584.
• [25] M. Sapir, Some group theory problems, Internat. J. Algebra Comput. 17 (2007), 1189–1214.
• [26] J. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), 551–565.
• [27] W. A. Stein et al., Sage Mathematics Software (Version 5.3), Sage Development Team (2012), available at http://www.sagemath.org.
• [28] W. Thurston, A norm for the homology of 3-manifolds, Mem. AMS 59 (1986), i–vi and 99–130.
• [29] A. Walker, wallop, computer program available at https://github.com/aldenwalker/wallop (accessed 17 March 2015).