an:02011913
Zbl 1034.20038
Levitt, Gilbert; Lustig, Martin
Irreducible automorphisms of \(F_n\) have north-south dynamics on compactified outer space
EN
J. Inst. Math. Jussieu 2, No. 1, 59-72 (2003).
00095252
2003
j
20F65 20E05 20E36 20E08 57M07
free groups; outer spaces; outer automorphisms; irreducible automorphisms; pseudo-Anosov maps; train tracks
Let \(F_n\) be the non-Abelian free group of rank \(n\). In their study of the outer automorphism group \(\text{Out}(F_n)\) of \(F_n\), Culler and Vogtmann defined a moduli space \(CV_n\) of marked graphs, called `outer space', which is finite dimensional, contractible and which has a spine which admits a discrete co-compact action with finite point stabilizers of \(\text{Out}(F_n)\). \textit{M. Bestvina} and \textit{M. Handel}, in their study of the automorphisms of \(F_n\), introduced in their paper [Ann. Math. (2) 135, No. 1, 1-51 (1992; Zbl 0757.57004)] an analogue of Thurston's pseudo-Anosov maps, and they called these maps ``irreducible automorphisms'' of \(F_n\).
In the paper under review, the authors study the action of an irreducible automorphism on the closure \(\overline{CV_n}\) of \(CV_n\). They prove that if \(\alpha\in\Aut(F_n)\) is irreducible with irreducible powers, then its action on \(\overline{CV_n}\) has north-south dynamics. In other words, there exist two points \([T^+]\) and \([T^-]\) in \(\partial CV_n\) such that \(\alpha^p([T])\to[T^+]\) as \(p\to\infty\) for all \([T]\not=[T^-]\) and \(\alpha^{-p}([T])\to[T^-]\) as \(p\to\infty\) for all \([T]\not=[T^+]\). This property is an analog of a property of the action of a pseudo-Anasov mapping class on Thurston's compactification of Teichm??ller space.
Athanase Papadopoulos (Strasbourg)
Zbl 0757.57004