Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems

Francaviglia, Stefano; Martino, Armando

doi:10.1090/tran/8535

Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems
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Trans. Amer. Math. Soc. 375 (2022), 2511-2551 Request permission

Abstract:

This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper [Trans. Amer. Math. Soc. 374 (2021), pp. 3215–3264)] we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity).

This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. Here, by the “level set” we intend to indicate the set of points displaced by at most some amount, rather than exactly some amount; this is sometimes called a “sub-level set”.

As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms.

Note: The two papers were originally packed together in the preprint (On the connectivity of level sets of automorphisms of free groups, with applications to decision problems, arXiv:1703.09945) We decided to split that paper following the recommendations of a referee.

References

Yael Algom-Kfir, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011), no. 4, 2181–2233. MR 2862155, DOI 10.2140/gt.2011.15.2181
Yael Algom-Kfir and Mladen Bestvina, Asymmetry of outer space, Geom. Dedicata 156 (2012), 81–92. MR 2863547, DOI 10.1007/s10711-011-9591-2
Mladen Bestvina, A Bers-like proof of the existence of train tracks for free group automorphisms, Fund. Math. 214 (2011), no. 1, 1–12. MR 2845630, DOI 10.4064/fm214-1-1
Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for $\textrm {Out}(F_n)$. II. A Kolchin type theorem, Ann. of Math. (2) 161 (2005), no. 1, 1–59. MR 2150382, DOI 10.4007/annals.2005.161.1
Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1–51. MR 1147956, DOI 10.2307/2946562
Mark Feighn and Michael Handel, Algorithmic constructions of relative train track maps and cts.
Mark Feighn and Michael Handel, The recognition theorem for $\textrm {Out}(F_n)$, Groups Geom. Dyn. 5 (2011), no. 1, 39–106. MR 2763779, DOI 10.4171/GGD/116
Stefano Francaviglia, Geodesic currents and length compactness for automorphisms of free groups, Trans. Amer. Math. Soc. 361 (2009), no. 1, 161–176. MR 2439402, DOI 10.1090/S0002-9947-08-04420-6
Stefano Francaviglia and Armando Martino, Stretching factors, metrics and train tracks for free products, Illinois J. Math. 59 (2015), no. 4, 859–899. MR 3628293
Stefano Francaviglia and Armando Martino, On the connectivity of level sets of automorphisms of free groups, with applications to decision problems, arXiv:1703.09945.
Stefano Francaviglia and Armando Martino, Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks, Trans. Amer. Math. Soc. 374 (2021), no. 5, 3215–3264. MR 4237947, DOI 10.1090/tran/8333
S. M. Gersten, Addendum: “On fixed points of certain automorphisms of free groups”, Proc. London Math. Soc. (3) 49 (1984), no. 2, 340–342. MR 748994, DOI 10.1112/plms/s3-49.2.340-s
S. M. Gersten, On fixed points of certain automorphisms of free groups, Proc. London Math. Soc. (3) 48 (1984), no. 1, 72–90. MR 721773, DOI 10.1112/plms/s3-48.1.72
Vincent Guirardel and Gilbert Levitt, The outer space of a free product, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 695–714. MR 2325317, DOI 10.1112/plms/pdl026
Michael Handel and Lee Mosher, The free splitting complex of a free group, I: hyperbolicity, Geom. Topol. 17 (2013), no. 3, 1581–1672. MR 3073931, DOI 10.2140/gt.2013.17.1581
Michael Handel and Lee Mosher, The free splitting complex of a free group, II: Loxodromic outer automorphisms, Trans. Amer. Math. Soc. 372 (2019), no. 6, 4053–4105. MR 4009387, DOI 10.1090/tran/7698
Camille Horbez, Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings, J. Topol. 9 (2016), no. 2, 401–450. MR 3509969, DOI 10.1112/jtopol/jtv045
Ilya Kapovich, Algorithmic detectability of iwip automorphisms, Bull. Lond. Math. Soc. 46 (2014), no. 2, 279–290. MR 3194747, DOI 10.1112/blms/bdt093
Ilya Kapovich, Detecting fully irreducible automorphisms: a polynomial time algorithm, Exp. Math. 28 (2019), no. 1, 24–38. With an appendix by Mark C. Bell. MR 3938575, DOI 10.1080/10586458.2017.1326326
Jérôme E. Los, On the conjugacy problem for automorphisms of free groups, Topology 35 (1996), no. 3, 779–808. With an addendum by the author. MR 1396778, DOI 10.1016/0040-9383(95)00035-6
Sebastian Meinert, The Lipschitz metric on deformation spaces of $G$-trees, Algebr. Geom. Topol. 15 (2015), no. 2, 987–1029. MR 3342683, DOI 10.2140/agt.2015.15.987
John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551–565. MR 695906, DOI 10.1007/BF02095993
E. Ventura, Fixed subgroups in free groups: a survey, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 231–255. MR 1922276, DOI 10.1090/conm/296/05077