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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.

MSC:
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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