Test elements and the retract theorem in hyperbolic groups
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 restriction \(\phi_\infty\) of \(\phi^N\) to \(\phi^\infty(G)\) is an automorphism. Then, if \(\phi(g)=g\) for a proper endomorphism, and \(\pi\) denotes the projection of \(\phi^N(G)\) to \(\phi_\infty(G)\), the composition \(\phi_\infty^{-N}\circ\pi\circ\phi^N\) is a retract from \(G\) to \(\phi^\infty(G)\), and the latter contains \(g\).
scientific article; zbMATH DE number 1497689