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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.