Mapping tori of free group automorphisms are coherent

Mark Feighn and Michael Handel

Review from Zentralblatt MATH:

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 $\bbfF$ has a basis ${\cal E}=\{e_i\mid i\in I\}$ and $\Phi$ is an injective endomorphism of $\bbfF$ then the mapping torus denoted $M(\Phi)$ is the group which has a presentation with generators ${\cal E}\cup\{t\}$ 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 $\bbfF$ 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 $\bbfF$, (ii) $r_j=ta_jt^{-1}w^{-1}_j$ for $w_j=\Phi(a_j)$ and $1\le j\le m$ and (iii) $\langle A,\phi(A)\rangle=\langle A,B\rangle$.

The methods are geometric. The main techniques are relative to Stallings folds [{\it J. R. Stallings}, Invent. Math. 71, 551-565 (1983; Zbl 0521.20013)]. The list of references contains 22 relative papers.

Reviewed by S.Andreadakis

Keywords: coherent groups; finitely generated subgroups; mapping tori; injective endomorphisms; HNN-extensions; presentations

Classification (MSC2000): 20E06 20E05 57M07 20E36 20F05

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