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Growth of intersection numbers for free group automorphisms. (English) Zbl 1209.20031
Let \(G\) be a group and \(T,T'\) two \(G\)-trees. V. Guirardel [in Ann. Sci. Éc. Norm. Supér. (4) 38, No. 6, 847-888 (2005; Zbl 1110.20019)] constructed a core \(\mathcal C(T\times T')\subset T\times T'\). The group \(G\) acts on the core \(\mathcal C(T\times T')\) and the intersection number between the two \(G\)-trees \(T,T'\), denoted by \(i(T,T')\), is defined to be the volume of the quotient \(\mathcal C(T\times T')/G\).
Let \(F_k\) be the free group of rank \(k\) and \(\mathcal{CV}_k\) be the Culler and Vogtmann’s outer space. If \(cv_k\) is the unprojectivized version of \(\mathcal{CV}_k\), \(T,T'\) two trees in \(cv_k\) and \(\varphi\) an outer automorphism of \(F_k\), in the present paper, it is given a method for computing the intersection number \(i(T,T')\) and it is proved that the asymptotics of \(n\to i(T,T'\varphi^n)\) do not depend on the trees \(T\) and \(T'\).
More precisely it is proved the Theorem: Suppose that \(\varphi\in\text{Out}(F_k)\) is fully irreducible with an expansion factor \(\lambda\) and \(T,T'\in cv_k\). Let \(T^+\) be the stable tree for \(\varphi\) and let \(\mu\) be the expansion factor of \(\varphi^{-1}\). Then we have either of the following.
(1) If \(T^+\) is geometric, then \(i(T,T'\varphi^n)\sim\lambda^n\).
(2) If \(T^+\) is nongeometric, then \(i(T,T'\varphi^n)\sim\lambda^n+\lambda^{n-1}\mu+\cdots+\lambda\mu^{n-1}+\mu^n\).
In the statement above \(\sim\) means quasi-isometry in the usual sense. For the terminology: stable tree, expansion factor, (non)geometric tree, we refer to the introduction of the paper and to the references quoted there.

20E36 Automorphisms of infinite groups
20E08 Groups acting on trees
20F65 Geometric group theory
20E05 Free nonabelian groups
20F69 Asymptotic properties of groups
57M07 Topological methods in group theory
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