Mapping tori of free group automorphisms are coherent. (English) Zbl 0938.20022

A group is coherent if its finitely generated subgroups are finitely presented. In this paper, the authors consider mapping tori of free group injective endomorphisms. The mapping torus of an injective endomorphism \(\Phi\) of a free group \(G\) is the HNN-extension \(G*_G\), where the bonding maps are the identity and \(\Phi\), i.e., the group generated by \(G\) and \(t\) and relators \(tgt^{-1}\Phi(g)^{-1}\), \(g\in G\). So if the free group \(\mathbb{F}\) has a basis \({\mathcal E}=\{e_i\mid i\in I\}\) and \(\Phi\) is an injective endomorphism of \(\mathbb{F}\) then the mapping torus denoted \(M(\Phi)\) is the group which has a presentation with generators \({\mathcal E}\cup\{t\}\) and relators \(\{te_it^{-1}\Phi(e_i)^{-1}\mid i\in I\}\). The main result states that the mapping torus \(M(\Phi)\) of an injective endomorphism \(\Phi\) of a (possibly infinite rank) free group is coherent. Furthermore, they prove that the finitely generated subgroups of \(M(\Phi)\) are also of finite type, i.e., they have a compact Eilenberg-MacLane space. The proof of the main theorem is carried out with the aid of the following (main proposition): If \(\Phi\) is an injective endomorphism of \(\mathbb{F}\) and if \(H\) is a finitely generated subgroup of \(M(\Phi)\) that contains \(t\), then \(H\) has a presentation of the form \(\langle t,A,B\mid C\rangle\) where (i) \(A=\{a_1,\dots,a_m\}\), \(B=\{b_1,\dots,b_r\}\), \(C=\{r_1,r_2,\dots,r_m\}\) are finite sets in \(\mathbb{F}\), (ii) \(r_j=ta_jt^{-1}w^{-1}_j\) for \(w_j=\Phi(a_j)\) and \(1\leq j\leq m\) and (iii) \(\langle A,\phi(A)\rangle=\langle A,B\rangle\).
The methods are geometric. The main techniques are relative to Stallings folds [J. R. Stallings, Invent. Math. 71, 551-565 (1983; Zbl 0521.20013)]. The list of references contains 22 relative papers.

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Examples of locally hyperbolic groups


20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E05 Free nonabelian groups
57M07 Topological methods in group theory
20E36 Automorphisms of infinite groups
20F05 Generators, relations, and presentations of groups


Zbl 0521.20013
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