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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.

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)

Citations:

Zbl 0757.57004
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