Test elements and the retract theorem in hyperbolic groups
type: Item
instance of: scholarly article
title: Test elements and the retract theorem in hyperbolic groups
zbMATH Open document ID: 0954.20020
author: J. O'Neill
author: Edward C. Turner
published in: The New York Journal of Mathematics
publication date: 2000-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/120756
full work available at URL: 6-7.html
review text: An element \(g\) of a group \(G\) is called a test element if for any endomorphism \(\phi\colon G\to G\), \(\phi(g)=g\) implies that \(\phi\) is an automorphism. Clearly, if \(g\) is in the image of a proper retract from \(G\) to \(G\), then \(g\) is not a test element. In previous work, the second author gave a kind of converse to this for free groups, by showing that in this case the test elements are precisely those elements that are not in the image of any proper retract [\textit{E.~C.~Turner}, Bull. Lond. Math. Soc. 28, No. 3, 255-263 (1996; Zbl 0852.20022)]. This is called the Retract Theorem. In this paper, the authors extend the Retract Theorem to a large class of hyperbolic groups (in the sense of Gromov), the torsion-free stably hyperbolic groups. The latter condition means the hyperbolic groups \(G\) with the property that if \(\phi\) is any endomorphism, there are arbitrarily large \(n\) for which \(\phi^n(G)\) is hyperbolic. Using an idea of G.~A.~Swarup, the authors give examples showing that there can be endomorphisms for which \(\phi^n(G)\) is nonhyperbolic for arbitrarily large \(n\), but for all known examples the images are eventually hyperbolic. The authors also extend the Retract Theorem to all Fuchsian groups, and show that every surface group other than \(\mathbb{Z}\times\mathbb{Z}\) contains test elements. Interestingly, the fundamental group \(\langle a,b\mid aba^{-1}=b^{-1}\rangle\) does not satisfy the conclusion of the Retract Theorem; the element \(b\) lies in no proper retract, but also is not a test element since it is fixed by the proper endomorphism \(\varphi(a)=a^3\), \(\varphi(b)=b\). The key idea in the extensions of the Retract Theorem is to show that in a stably hyperbolic group or a Fuchsian group, the intersection \(\phi^\infty(G)\) of the \(\phi^n(G)\) is a free factor of \(\phi^N(G)\) for some \(N\). This requires quite a bit of argument. Also, using a result of Z.~Sela in the stably hyperbolic case, it is shown that the res…
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20F38
zbMATH DE Number: 1497689
zbMATH Keywords: endomorphisms
zbMATH Keywords: automorphisms
zbMATH Keywords: test elements
zbMATH Keywords: surface groups
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: Hopfian groups
zbMATH Keywords: co-Hopfian groups
zbMATH Keywords: retract theorem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1497689