zbMATH — the first resource for mathematics

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

20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.