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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)
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