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

Author:
Mihalis Sykiotis

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

MSC (2000):
Primary 20E36, 20E08, 20E06

Published electronically:
November 12, 2003

MathSciNet review:
2048523

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**1.**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**, 10.1007/978-1-4612-3142-4_3**2.**Mladen Bestvina and Michael Handel,*Train tracks and automorphisms of free groups*, Ann. of Math. (2)**135**(1992), no. 1, 1–51. MR**1147956**, 10.2307/2946562**3.**Mladen Bestvina, Mark Feighn, and Michael Handel,*The Tits alternative for 𝑂𝑢𝑡(𝐹_{𝑛}). I. Dynamics of exponentially-growing automorphisms*, Ann. of Math. (2)**151**(2000), no. 2, 517–623. MR**1765705**, 10.2307/121043**4.**M. Bestvina, M. Feighn and M. Handel,*The Tits alternative for II: A Kolchin type theorem*, preprint.**5.**M. Bestvina, M. Feighn and M. Handel,*The Tits alternative for III: Solvable subgroups of are virtually abelian*, preprint.**6.**Daniel E. Cohen,*Combinatorial group theory: a topological approach*, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989. MR**1020297****7.**Donald J. Collins and Edward C. Turner,*Free product fixed points*, J. London Math. Soc. (2)**38**(1988), no. 1, 67–76. MR**949082**, 10.1112/jlms/s2-38.1.67**8.**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**, 10.1307/mmj/1029005072**9.**Warren Dicks and M. J. Dunwoody,*Groups acting on graphs*, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR**1001965****10.**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****11.**S. M. Gersten,*Fixed points of automorphisms of free groups*, Adv. in Math.**64**(1987), no. 1, 51–85. MR**879856**, 10.1016/0001-8708(87)90004-1**12.**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**, 10.1112/blms/18.5.468**13.**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**, 10.1007/BF01196349**14.**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504****15.**M. Sykiotis,*Fixed points of symmetric endomorphisms of groups*, Internat. J. Algebra Comput. (5)**12**(2002), 737-745.

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Additional Information

**Mihalis Sykiotis**

Affiliation:
Department of Mathematics, University of Athens, Athens 15784, Greece

Address at time of publication:
Amalthias 18, Larisa 41222, Greece

Email:
msikiot@cc.uoa.gr

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

Received by editor(s):
July 24, 2002

Received by editor(s) in revised form:
April 17, 2003

Published electronically:
November 12, 2003

Article copyright:
© Copyright 2003
American Mathematical Society