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Axes in outer space. (English) Zbl 1238.57002
Mem. Am. Math. Soc. 1004, v, 104 p. (2011).
The authors develop a notion of axis-bundle for the action of a nongeometric, fully irreducible outer automorphism on outer space. The Culler-Vogtmann outer space \(X_r\) of a finite rank free group \(F_r\) has no natural metric. Hence there does not seem to be a natural notion of an axis for a fully irreducible \(\phi \in Out(F_r)\). Nevertheless, it is shown in the paper under review that the axes for \(\phi\) fit into an axis bundle \(A_\phi\) satisfying the following topological properties:
1.
\(A_\phi\) is a closed subset of \(X_r\) proper homotopy equivalent to a line,
2.
\(A_\phi\) is invariant under \(\phi\),
3.
The two boundary points of \(A_\phi\) on the boundary of outer space correspond to the repeller and attractor of the source-sink action of \(\phi\), i.e. the negative end of \(A_\phi\) limits in compactified outer space on the repeller, and the positive end limits on the attractor.
4.
\(A_\phi\) depends naturally on the repeller and attractor, i.e. for any nongeometric, fully irreducible \(\phi,\phi' \in Out(F_r)\) and any \(\psi \in Out(F_r)\), if \(\psi\) takes the repeller-attractor pair of \(\phi\) to the repeller-attractor pair of \(\phi'\), then \(\psi\) takes \(A_\phi\) to \(A_{\phi'}\). It follows that \(A_\phi\) is invariant under \(\phi\) and that \(\phi^n\) and \(\phi\) have the same axis bundle for all \(n>0\).
The authors propose various definitions for \(A_\phi\), motivated by various aspects of the theory of outer automorphisms, and by analogy with various properties of Teichmüller geodesics. The main thrust of the paper is to prove the equivalence between these definitions.
Reviewer: Mahan Mj (Howrah)

MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20F65 Geometric group theory
57M07 Topological methods in group theory
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