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Fixed Subgroups of Homomorphisms of Free Groups

Fixed Subgroups of Homomorphisms of Free Groups FIXED SUBGROUPS OF HOMOMORPHISMS OF FREE GROUPS RICHAR D Z. GOLDSTEIN AND EDWARD C. TURNER THEOREM. If H a F is a finitely generated subgroup of a free group and <f>\H' -> F is a homomorphism, then , Hx ((f>) = {w m = w} is finitely generated. This theorem improves results of [1, 2, 3, 4, 5, 6, 7, 9,10]. In particular, we have proven the conjecture of Stallings [9]. COROLLARY. If <j>, H:F-> F (F finitely generated) are a homomorphism and a monomorphism respectively, then is finitely generated. Choose a basis for F and let N = {a ..., a } be a Nielsen reduced basis for H. 15 fc If OLEN = {af,..., a^}, then a can be written as a = i w r , m # 0 , so that in a a a a ± ± any product of elements in N , the m never cancel [8, p . 7]. Fo r any w e F and <xeN , we say there is large cancellation in the product w • a if i m cancels. That N is Nielsen a a reduced means that for any w, there is large cancellation in http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Fixed Subgroups of Homomorphisms of Free Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/18.5.468
Publisher site
See Article on Publisher Site

Abstract

FIXED SUBGROUPS OF HOMOMORPHISMS OF FREE GROUPS RICHAR D Z. GOLDSTEIN AND EDWARD C. TURNER THEOREM. If H a F is a finitely generated subgroup of a free group and <f>\H' -> F is a homomorphism, then , Hx ((f>) = {w m = w} is finitely generated. This theorem improves results of [1, 2, 3, 4, 5, 6, 7, 9,10]. In particular, we have proven the conjecture of Stallings [9]. COROLLARY. If <j>, H:F-> F (F finitely generated) are a homomorphism and a monomorphism respectively, then is finitely generated. Choose a basis for F and let N = {a ..., a } be a Nielsen reduced basis for H. 15 fc If OLEN = {af,..., a^}, then a can be written as a = i w r , m # 0 , so that in a a a a ± ± any product of elements in N , the m never cancel [8, p . 7]. Fo r any w e F and <xeN , we say there is large cancellation in the product w • a if i m cancels. That N is Nielsen a a reduced means that for any w, there is large cancellation in

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Sep 1, 1986

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