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Reactions to proofs of Gersten's Fixed Point Theorem

1.
Fixed subgroups of automorphisms of free by finite groups, Arch. Math. 48 (1987), 25-30.

2.
On graphs realizing automorphisms of free groups, Proc. Amer. Math. Soc. 107 (1989), 573-575.

3.
On the rank of the fixed point set of automorphisms of free groups (with W. Imrich and E. C. Turner), Circles and Rays (G. Hahn et al., eds) Kluwer, Dordrecht 1989.



SUMMARY:


Gersten was the first to prove that the fixed point subgroup of an automorphism of Fn is finitely generated [G]. The existence of a specific graph realizing a given automorphism (``Gersten bridge'') was reproved by several autors; the shortest proof is in my article [2]. Analyzing the Goldstein-Turner's proof of Gersten's Theorem [GT], we obtain in [3] a bound for the rank of the fixed-point subgroup. (The same bound was also proved in [CL].) I also noticed that Cooper's proof of the Gersten's Theorem [C] generalizes to virtually free groups [1]. Later, Paulin showed that, in fact, the generalization goes on to cover the whole class of hyperbolic groups [P].



REFERENCES:


[C]
D. Cooper, Automorphisms of free groups have finitely generated fixed point set, J. Algebra 111 (1987), 453-456.

[M]
M. Cohen and M. Lustig, On the dynamics and the fixed subgroup of a free group automorphism, Inventiones Math. 96 (1989), 613-638.

[G]
S. M. Gersten, Fixed points of automorphisms of free groups, Adv. in Math. 64 (1987), 51-85.

[GT]
R. Z. Goldstein and E. C. Turner, Fixed subgroups of homomorphisms of free groups, Bull. London Math. Soc. 18 (1986), 468-470.

[P]
F. Paulin, Points fixes des automorphismes de groupes hyperboliques, Ann. Inst. Fourier 39 (1989), 651-662.


next up previous
Next: Enumeration of quadratic words Up: SUMMARY OF RESULTS Previous: Kervaire-Laudenbach Conjecture

2000-03-16