The group fixed by a family of injective endomorphisms of a free group.

*(English)*Zbl 0845.20018
Contemporary Mathematics. 195. Providence, RI: American Mathematical Society (AMS). 81 p. (1996).

The purpose of this monograph is to give a selfcontained and purely algebraic detailed proof of the M. Bestvina-M. Handel Theorem [Ann. Math., II. Ser. 135, No. 1, 1-51 (1992; Zbl 0757.57004)] that the rank of the subgroup of the fixed elements of an automorphism of a free group \(F\) of rank \(n\) is at most \(n\).

The authors use the theory and the language of groupoids adapted conveniently to their purposes. They define (among others) a new concept, the concept of an inert subgroup as a subgroup \(H\) of a free group \(F\) such that \(r(H\cap K)\leq r(K)\) for any subgroup \(K\) of \(F\). Their proof gives something more general, because they prove that if \(B\) is a set of injective endomorphisms of \(F\) (\(F\) of finite rank), then \(\text{Fix}(B)\) is inert in \(F\) and so in particular \(r(\text{Fix}(B))\leq r(F)\).

The exposition seems condensed for the non-specialist, but it is clear in spite of the introduction of numerous terms and symbolisms. They conclude with a set of seven problems, concerning primarily endomorphisms of free groups, arising from their considerations.

The authors use the theory and the language of groupoids adapted conveniently to their purposes. They define (among others) a new concept, the concept of an inert subgroup as a subgroup \(H\) of a free group \(F\) such that \(r(H\cap K)\leq r(K)\) for any subgroup \(K\) of \(F\). Their proof gives something more general, because they prove that if \(B\) is a set of injective endomorphisms of \(F\) (\(F\) of finite rank), then \(\text{Fix}(B)\) is inert in \(F\) and so in particular \(r(\text{Fix}(B))\leq r(F)\).

The exposition seems condensed for the non-specialist, but it is clear in spite of the introduction of numerous terms and symbolisms. They conclude with a set of seven problems, concerning primarily endomorphisms of free groups, arising from their considerations.

Reviewer: S.Andreadakis (Athens)

##### MSC:

20E05 | Free nonabelian groups |

20E36 | Automorphisms of infinite groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20E08 | Groups acting on trees |

20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |