Intersections of automorphism fixed subgroups in the free group of rank three

Martino, Armando¹

¹Centre de Recerca Matemetica, Apartat 50, E-08193 Bellaterra, Spain

Algebraic and Geometric Topology, Tome 4 (2004) no. 1, pp. 177-198

Résumé

We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.

Détail

DOI : 10.2140/agt.2004.4.177

Keywords: free group, automorphism, fixed subgroup

Bibliographie

[1] G M Bergman, Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups, Trans. Amer. Math. Soc. 351 (1999) 1531

[2] M Bestvina, M Feighn, M Handel, The Tits alternative for $\mathrm{Out}(F_n)$ I: Dynamics of exponentially-growing automorphisms, Ann. of Math. $(2)$ 151 (2000) 517

[3] M Bestvina, M Feighn, M Handel, The Tits alternative for $\mathrm{Out}(F_n)$ II: A Kolchin type theorem, Ann. of Math. $(2)$ 161 (2005) 1

[4] M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. $(2)$ 135 (1992) 1

[5] D J Collins, E C Turner, All automorphisms of free groups with maximal rank fixed subgroups, Math. Proc. Cambridge Philos. Soc. 119 (1996) 615

[6] W Dicks, E Ventura, The group fixed by a family of injective endomorphisms of a free group, Contemporary Mathematics 195, American Mathematical Society (1996)

[7] J L Dyer, G P Scott, Periodic automorphisms of free groups, Comm. Algebra 3 (1975) 195

[8] W Imrich, E C Turner, Endomorphisms of free groups and their fixed points, Math. Proc. Cambridge Philos. Soc. 105 (1989) 421

[9] I Kapovich, A Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002) 608

[10] R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer (1977)

[11] W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers, New York-London-Sydney (1966)

[12] A Martino, E Ventura, A description of auto-fixed subgroups in a free group, Topology 43 (2004) 1133

[13] A Martino, E Ventura, Fixed subgroups are compressed in free groups, Comm. Algebra 32 (2004) 3921

[14] A Martino, E Ventura, On automorphism-fixed subgroups of a free group, J. Algebra 230 (2000) 596

[15] J R Stallings, Whitehead graphs on handlebodies, from: "Geometric group theory down under (Canberra, 1996)", de Gruyter (1999) 317

[16] E C Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996) 255

[17] E Ventura, On fixed subgroups of maximal rank, Comm. Algebra 25 (1997) 3361

[18] J H C Whitehead, On certain sets of elements in a free group, Proc. London. Math. Soc. 4148