Stable representatives for symmetric automorphisms of groups and the general form of the Scott conjecture

Sykiotis, Mihalis

doi:10.1090/S0002-9947-03-03385-3

Stable representatives for symmetric automorphisms of groups and the general form of the Scott conjecture
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by Mihalis Sykiotis

Trans. Amer. Math. Soc. 356 (2004), 2405-2441

DOI: https://doi.org/10.1090/S0002-9947-03-03385-3

Published electronically: November 12, 2003

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Abstract:

Let $G$ be a group acting on a tree $X$ such that all edge stabilizers are finite. We extend Bestvina-Handel’s theory of train tracks for automorphisms of free groups to automorphisms of $G$ which permute vertex stabilizers. Using this extension we show that there is an upper bound depending only on $G$ for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of $G$.

References

Hyman Bass, Group actions on non-Archimedean trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 69–131. MR 1105330, DOI 10.1007/978-1-4612-3142-4_{3}
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
Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for $\textrm {Out}(F_n)$. I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517–623. MR 1765705, DOI 10.2307/121043
M. Bestvina, M. Feighn and M. Handel, The Tits alternative for $Out(F_{n})$ II: A Kolchin type theorem, preprint.
M. Bestvina, M. Feighn and M. Handel, The Tits alternative for $Out(F_{n})$ III: Solvable subgroups of $Out(F_{n})$ are virtually abelian, preprint.
Daniel E. Cohen, Combinatorial group theory: a topological approach, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989. MR 1020297, DOI 10.1017/CBO9780511565878
Donald J. Collins and Edward C. Turner, Free product fixed points, J. London Math. Soc. (2) 38 (1988), no. 1, 67–76. MR 949082, DOI 10.1112/jlms/s2-38.1.67
D. J. Collins and E. C. Turner, Efficient representatives for automorphisms of free products, Michigan Math. J. 41 (1994), no. 3, 443–464. MR 1297701, DOI 10.1307/mmj/1029005072
Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR 1001965
Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, Contemporary Mathematics, vol. 195, American Mathematical Society, Providence, RI, 1996. MR 1385923, DOI 10.1090/conm/195
S. M. Gersten, Fixed points of automorphisms of free groups, Adv. in Math. 64 (1987), no. 1, 51–85. MR 879856, DOI 10.1016/0001-8708(87)90004-1
Richard Z. Goldstein and Edward C. Turner, Fixed subgroups of homomorphisms of free groups, Bull. London Math. Soc. 18 (1986), no. 5, 468–470. MR 847985, DOI 10.1112/blms/18.5.468
Sava Krstić, Fixed subgroups of automorphisms of free by finite groups: an extension of Cooper’s proof, Arch. Math. (Basel) 48 (1987), no. 1, 25–30. MR 878002, DOI 10.1007/BF01196349
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
M. Sykiotis, Fixed points of symmetric endomorphisms of groups, Internat. J. Algebra Comput. (5) 12 (2002), 737-745.