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Geodesic currents and length compactness for automorphisms of free groups. (English) Zbl 1166.20032
Let $$F$$ be a free group of finite rank $$k\geq 2$$ and $$\vartheta F$$ its hyperbolic boundary, which can be viewed as the space of right-infinite, freely reduced, words in an alphabet of $$F$$. The boundary is endowed with the Cantor-set topology. Let $$\vartheta^2F=\{(x,y)\mid x,y\in\vartheta F$$, $$x\neq y\}$$. $$\vartheta^2F$$ can be identified with the set of oriented bi-infinite geodesics in the Cayley graph of $$F$$. The base-ball of $$\vartheta^2F$$ is the set of geodesics passing through 1.
A geodesic current on $$F$$ is a non-negative locally finite Borel measure on $$\vartheta^2F$$ that is $$F$$-invariant. The length $$L(\eta)$$ of a current $$\eta$$ is the measure $$\eta(B)$$ of the base-ball of $$\vartheta^2F$$.
If $$\varphi$$ is an automorphism of $$F$$, then it is extended to a homeomorphism of $$\vartheta F$$ (which is denoted again by $$\varphi$$) [D. Cooper, J. Algebra 111, 453-456 (1987; Zbl 0628.20029)]. This gives rise to an action of $$\varphi$$ on the space of the geodesic currents on $$F$$: Let $$\eta$$ be a geodesic current, define $$\varphi\eta$$ by setting $$\varphi\eta(S)=\eta(\varphi^{-1}(S))$$ for a Borel subset $$S\subseteq\vartheta^2F$$. Then it was proved [by I. Kapovich, Topological and asymptotic aspects of group theory. Contemp. Math. 394, 149-176 (2006; Zbl 1110.20034)] that $$\varphi\eta$$ is a geodesic current on $$F$$.
Let $$T\colon\vartheta F\to\vartheta F$$ be the shift operator deleting the first letter of one right-infinite word. The space of frequency measures is the set of finite-mass $$T$$-invariant non-negative Borel measures on $$\vartheta F$$.
There is a natural homeomorphism from the space of frequency measures to the space of geodesic currents (Theorem 3.4 in the paper or in [I. Kapovich, Exp. Math. 16, No. 1, 67-76 (2007; Zbl 1158.20014)]), therefore the action of an automorphism of $$F$$ on the space of the geodesic currents on $$F$$ induces an action of this automorphism on the frequency measures.
If we pass to the uniform current $$\eta_A$$ (see Definition 3.5 in the paper), then the length of the automorphism $$\varphi$$ is defined as the length of the image of the uniform current, that is, $$L(\varphi)=L(\varphi\eta_A)=\eta_A(\varphi^{-1}(B))$$.
Now we can state the main result of the paper: For a sequence $$\varphi_n$$ of automorphisms of $$F$$, if there is a word $$w$$ such that the cyclically reduced length of $$\varphi_n(w)$$ goes to $$\infty$$, then $$L(\varphi_n)\to\infty$$.

##### MSC:
 20F65 Geometric group theory 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups
##### Keywords:
automorphisms; free groups; geodesic currents
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##### References:
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