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

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

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.**H. Bass,*Group actions on non-archimedean trees*, Arboreal group theory (R.C. Alperin, ed.), M.S.R.I. publ. vol. 19, Springer-Verlag, New York, (1991), pp. 69-131. MR**93d:57003****2.**M. Bestvina and M. Handel,*Train tracks and automorphisms of free groups*, Ann. of Math. (2)**135**(1992), 1-51. MR**92m:20017****3.**M. Bestvina, M. Feighn and M. Handel,*The Tits alternative for I: Dynamics of exponentially -growing automorphisms*, Ann. of Math. (2)**151**(2000), 517-623. MR**2002a:20034****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.**D. E. Cohen,*Combinatorial Group Theory: a topological approach*, London Math. Soc. Stud. Texts 14, Cambridge Univ. Press, Cambridge, 1989. MR**91d:20001****7.**D. J. Collins and E. C. Turner,*Free product fixed points*, J. London Math. Soc. (2)**38**(1988), 67-76. MR**89h:20037****8.**D. J. Collins and E. C. Turner,*Efficient representatives for automorphisms of free products*, Michigan Math. J.**41**(1994), 443-464. MR**95k:20039****9.**W. Dicks and M. J. Dunwoody,*Groups acting on graphs*, Cambridge Univ. Press, 1989. MR**91b:20001****10.**W. Dicks and E. Ventura,*The group fixed by a family of injective endomorphisms of a free group*, Contemp. Math.**195**(1996), 1-81.MR**97h:20030****11.**S. Gersten,*Fixed points of automorphisms of free groups*, Adv. in Math.**64**(1987), 51-85. MR**88f:20042****12.**R. Z. Goldstein and E. C. Turner,*Fixed subgroups of homomorphisms of free groups*, Bull. London Math. Soc.**18**(1986), 468-470.MR**87m:20096****13.**S. Krstic,*Fixed subgroups of automorphisms of free by finite groups: an extension of Cooper's proof*, Arch. Math. (Basel)**48**(1987), 25-30.MR**88d:20044****14.**J.-P. Serre,*Trees*, Springer-Verlag, New York, 1980. MR**82c:20083****15.**M. Sykiotis,*Fixed points of symmetric endomorphisms of groups*, Internat. J. Algebra Comput. (5)**12**(2002), 737-745.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20E36,
20E08,
20E06

Retrieve articles in all journals with MSC (2000): 20E36, 20E08, 20E06

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:
https://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