A McCool Whitehead type theorem for finitely generated subgroups of Out(F n )
Annales Henri Lebesgue, Volume 6 (2023), pp. 65-94.

Metadata

Keywords free group, geometric group theory

Abstract

S. Gersten announced an algorithm that takes as input two finite sequences K =(K 1 ,,K N ) and K =(K 1 ,,K N ) of conjugacy classes of finitely generated subgroups of F n and outputs:

  • (1) YES or NO depending on whether or not there is an element θOut(F n ) such that θ(K )=K together with one such θ if it exists and
  • (2) a finite presentation for the subgroup of Out(F n ) fixing K .

S. Kalajdžievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler–Vogtmann’s Outer space. New results include that the subgroup of Out(F n ) fixing K is of type VF, an equivariant version of these results, an application, and a unified approach to such questions.


References

[AFV08] Armstrong, Heather; Forrest, Bradley; Vogtmann, Karen A presentation for Aut (F n ), J. Group Theory, Volume 11 (2008) no. 2, pp. 267-276 | DOI | MR | Zbl

[BBM07] Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan Dimension of the Torelli group for Out (F n ), Invent. Math., Volume 170 (2007) no. 1, pp. 1-32 | DOI | MR | Zbl

[Bes88] Bestvina, Mladen Degenerations of the hyperbolic space, Duke Math. J., Volume 56 (1988) no. 1, pp. 143-161 | DOI | MR | Zbl

[BF95] Bestvina, Mladen; Feighn, Mark Stable actions of groups on real trees, Invent. Math., Volume 121 (1995) no. 2, pp. 287-321 | DOI | MR | Zbl

[Bro84] Brown, Kenneth S. Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra, Volume 32 (1984) no. 1, pp. 1-10 | DOI | MR | Zbl

[CL99] Cohen, Marshall M.; Lustig, Martin The conjugacy problem for Dehn twist automorphisms of free groups, Comment. Math. Helv., Volume 74 (1999) no. 2, pp. 179-200 | DOI | MR | Zbl

[Cla05] Clay, Matt Contractibility of deformation spaces of G-trees, Algebr. Geom. Topol., Volume 5 (2005), pp. 1481-1503 | DOI | MR | Zbl

[Cul84] Culler, Marc Finite groups of outer automorphisms a free group, Contributions to group theory (Contemporary Mathematics), Volume 33, American Mathematical Society (1984), pp. 197-207 | DOI | MR | Zbl

[CV86] Culler, Marc; Vogtmann, Karen Moduli of graphs and automorphisms of free groups, Invent. Math., Volume 84 (1986) no. 1, pp. 91-119 | DOI | MR | Zbl

[DS99] Dunwoody, Martin J.; Sageev, M. E. JSJ-splittings for finitely presented groups over slender groups, Invent. Math., Volume 135 (1999) no. 1, pp. 25-44 | DOI | MR | Zbl

[FH99] Feighn, Mark; Handel, Michael Mapping tori of free group automorphisms are coherent, Ann. Math., Volume 149 (1999) no. 3, pp. 1061-1077 | DOI | MR | Zbl

[FH19] Feighn, Mark; Handel, Michael The conjugacy problem for UPG elements of Out (F n ) (2019) (https://arxiv.org/abs/1906.04147)

[For02] Forester, Max Deformation and rigidity of simplicial group actions on trees, Geom. Topol., Volume 6 (2002), pp. 219-267 | DOI | MR | Zbl

[FP06] Fujiwara, Koji; Papasoglu, Panos JSJ-decompositions of finitely presented groups and complexes of groups, Geom. Funct. Anal., Volume 16 (2006) no. 1, pp. 70-125 | DOI | MR | Zbl

[Ger84] Gersten, S. M. On Whitehead’s algorithm, Bull. Am. Math. Soc., Volume 10 (1984) no. 2, pp. 281-284 | DOI | MR | Zbl

[GL07] Guirardel, Vincent; Levitt, Gilbert Deformation spaces of trees, Groups Geom. Dyn., Volume 1 (2007) no. 2, pp. 135-181 | DOI | MR | Zbl

[GL15] Guirardel, Vincent; Levitt, Gilbert McCool groups of toral relatively hyperbolic groups, Algebr. Geom. Topol., Volume 15 (2015) no. 6, pp. 3485-3534 | DOI | MR | Zbl

[GL17] Guirardel, Vincent; Levitt, Gilbert JSJ decompositions of groups, Astérisque, Société Mathématique de France, 2017 no. 395 | MR | Zbl

[GL21] Guirardel, Vincent; Levitt, Gilbert Random subgroups, automorphisms, splittings, Ann. Inst. Fourier, Volume 71 (2021) no. 4, pp. 1363-1391 | DOI | Numdam | MR | Zbl

[GLP94] Gaboriau, Damien; Levitt, Gilbert; Paulin, Frédéric Pseudogroups of isometries of R and Rips’ theorem on free actions on R-trees, Isr. J. Math., Volume 87 (1994) no. 1-3, pp. 403-428 | DOI | MR | Zbl

[Gro75] Grossman, Edna K. On the residual finiteness of certain mapping class groups, J. Lond. Math. Soc., Volume 9 (1974/75), pp. 160-164 | DOI | MR | Zbl

[Kal92] Kalajdžievski, Sašo Automorphism group of a free group: centralizers and stabilizers, J. Algebra, Volume 150 (1992) no. 2, pp. 435-502 | DOI | MR | Zbl

[KLV01] Krstić, Sava; Lustig, Martin; Vogtmann, Karen An equivariant Whitehead algorithm and conjugacy for roots of Dehn twist automorphisms, Proc. Edinb. Math. Soc., II. Ser., Volume 44 (2001) no. 1, pp. 117-141 | DOI | MR | Zbl

[KM97] Krstić, Sava; McCool, James The non-finite presentability of IA (F 3 ) and GL 2 (Z[t,t -1 ]), Invent. Math., Volume 129 (1997) no. 3, pp. 595-606 | DOI | MR | Zbl

[KV93] Krstić, Sava; Vogtmann, Karen Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv., Volume 68 (1993) no. 2, pp. 216-262 | DOI | MR | Zbl

[Lev] Levitt, Gilbert McCool groups and stabilizers on the boundary of outer space (joint work with Vincent Guirardel, research announcement, http://people.maths.ox.ac.uk/drutu/conference/levitt.pdf)

[McC75] McCool, James Some finitely presented subgroups of the automorphism group of a free group, J. Algebra, Volume 35 (1975), pp. 205-213 | DOI | MR | Zbl

[Pau88] Paulin, Frédéric Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math., Volume 94 (1988) no. 1, pp. 53-80 | DOI | MR | Zbl

[RS97] Rips, Eliyahu; Sela, Zlil Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. Math., Volume 146 (1997) no. 1, pp. 53-109 | DOI | MR | Zbl

[Sta83] Stallings, John R. Topology of finite graphs, Invent. Math., Volume 71 (1983), pp. 551-565 | DOI | MR | Zbl

[SV87] Smillie, John; Vogtmann, Karen Automorphisms of graphs, p-subgroups of Out (F n ) and the Euler characteristic of Out (F n ), J. Pure Appl. Algebra, Volume 49 (1987) no. 1-2, pp. 187-200 | DOI | MR | Zbl

[Vog17] Vogtmann, Karen Contractibility of outer space: reprise, Hyperbolic geometry and geometric group theory. Proceedings of the 7th Seasonal Institute of the Mathematical Society of Japan (MSJ-SI), Tokyo, Japan, July 30 – August 5, 2014 (Advanced Studies in Pure Mathematics), Volume 73, Mathematical Society of Japan, 2017, pp. 265-280 | DOI | MR | Zbl

[Whi36a] Whitehead, John H. C. On certain sets of elements in a free group, Proc. Lond. Math. Soc., Volume 41 (1936), pp. 48-56 | DOI | MR | Zbl

[Whi36b] Whitehead, John H. C. On equivalent sets of elements in a free group, Ann. Math., Volume 37 (1936) no. 4, pp. 782-800 | DOI | MR | Zbl

[Zim81] Zimmermann, Bruno Über Homöomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv., Volume 56 (1981) no. 3, pp. 474-486 | DOI | MR | Zbl