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Monomorphisms of finitely generated free groups have finitely generated equalizers. (English) Zbl 0582.20023
If \(\phi\),\(\psi\) : \(G\to H\) are homomorphisms of groups, then the subgroup \(Eq(\phi,\psi)=\{x|\) \(\phi (x)=\psi (x)\}\) of G is called the equalizer of \(\phi\) and \(\psi\). The main result, which takes care of a conjecture of Stallings, is the following: If \(\phi\) and \(\psi\) are monomorphisms and G is a finitely generated free group, then Eq(\(\phi\),\(\psi)\) is finitely generated. The authors use graphical methods and prove all the main results in the 3-dimensional Whitehead model. They mention that J. Stallings [Graphical Theory of automorphisms of Free Groups, Proc. Alto Conf. Comb. Group Theory] and D. Cooper [Automorphisms of free groups have f.g. fixed point sets (preprint)] have also given a proof of this main result.
Reviewer: S.Andreadakis

MSC:
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
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References:
[1] Cooper, D.: Automorphisms of free groups have f.g. fixed point sets. (Preprint)
[2] Gersten, S.: Fixed points of automorphisms of free groups. (Preprint) · Zbl 0616.20014
[3] Goldstein, R.Z., Turner, E.C.: Automorphisms of free groups and their fixed points. Invent Math.78, 1-12 (1984) · Zbl 0548.20016 · doi:10.1007/BF01388713
[4] Hoare, A.H.M.: On automorphisms of free groups. (Preprint) · Zbl 0636.20019
[5] Stallings, J.: Topology of finite graphs. Invent Math.71, 551-565 (1983) · Zbl 0521.20013 · doi:10.1007/BF02095993
[6] Stallings, J.: Graphical Theory of automorphisms of Free groups. Proc. Alto Conf. Combin. Group Theory · Zbl 0619.20011
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