IRREDUCIBLE AUTOMORPHISMS OF $F_{n}$ HAVE NORTH–SOUTH DYNAMICS ON COMPACTIFIED OUTER SPACE

We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The...

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Published in:Journal of the Institute of Mathematics of Jussieu 2003-01, Vol.2 (1), p.59-72
Main Authors: Levitt, Gilbert, Lustig, Martin
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description We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point $Q(X)$ to any end $X\in\partial F_n$, given an action of $F_n$ on an $\bm{R}$-tree $T$ with trivial arc stabilizers; this point $Q(X)$ may be in $T$, or in its metric completion, or in its boundary. AMS 2000 Mathematics subject classification: Primary 20F65; 20E05; 20E08
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title IRREDUCIBLE AUTOMORPHISMS OF $F_{n}$ HAVE NORTH–SOUTH DYNAMICS ON COMPACTIFIED OUTER SPACE
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