The Group Fixed by a Family of Injective Endomorphisms of a Free Group (Contemporary Mathematics) - Softcover
Synopsis
This monograph contains a proof of the Bestvina-Handel Theorem (for any automorphism of a free group of rank $n$, the fixed group has rank at most $n$) that to date has not been available in book form. The account is self-contained, simplified, purely algebraic, and extends the results to an arbitrary family of injective endomorphisms. Let $F$ be a finitely generated free group, let $\phi$ be an injective endomorphism of $F$, and let $S$ be a family of injective endomorphisms of $F$.By using the Bestvina-Handel argument with graph pullback techniques of J. R. Stallings, the authors show that, for any subgroup $H$ of $F$, the rank of the intersection $H\cap \mathrm {Fix}(\phi)$ is at most the rank of $H$. They deduce that the rank of the free subgroup which consists of the elements of $F$ fixed by every element of $S$ is at most the rank of $F$. The topological proof by Bestvina-Handel is translated into the language of groupoids, and many details previously left to the reader are meticulously verified in this text.
"synopsis" may belong to another edition of this title.
Search results for The Group Fixed by a Family of Injective Endomorphisms...
Seller Image
The Group Fixed by a Family of Injective Endomorphisms of a Free Group (Contemporary Mathematics)
Published by
American Mathematical Society, 1996
ISBN 10: 0821805649
ISBN 13: 9780821805640
Used
Soft cover
Seller: Leopolis, Kraków, Poland
Seller rating 5 out of 5 stars
Soft cover. Condition: Fine. 8vo (25 cm), IX, 83 pp. Publisher's laminated wrappers (minor shelf-wear). A self-contained research monograph presenting the first book-length proof of the Bestvina-Handel theorem, establishing that for any automorphism of a free group of rank n, the fixed subgroup has rank at most n, and extending the result to arbitrary families of injective endomorphisms. The authors reformulate the original topological argument in the language of groupoids, incorporating Stallings's graph pullback techniques to show that for any finitely generated free group F, injective endomorphism ?, and subgroup H of F, the rank of H ? Fix(?) does not exceed the rank of H; from this it follows that the subgroup fixed by a family S of injective endomorphisms has rank at most that of F. Four chapters treat groupoids and abstract maps of graphs, measuring devices including the Perron-Frobenius theorem and metric graphs, properties of the basic operations (collapsing, subdividing, folding), and minimal representatives and fixed subgroupoids, followed by open problems, bibliography, and index. A concise but influential contribution to combinatorial and geometric group theory. Seller Inventory # 009827