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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:
 [1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0957.49001 [2] Francis Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), no. 1, 71 – 158 (French). · Zbl 0671.57008 [3] Daryl Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), no. 2, 453 – 456. · Zbl 0628.20029 [4] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 [5] Ilya Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 939 – 969. · Zbl 1110.20031 [6] Ilya Kapovich, Clusters, currents, and Whitehead’s algorithm, Experiment. Math. 16 (2007), no. 1, 67 – 76. · Zbl 1158.20014 [7] Ilya Kapovich, Currents on free groups, Topological and asymptotic aspects of group theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 149 – 176. · Zbl 1110.20034 [8] Vadim Kaimanovich, Ilya Kapovich, and Paul Shupp. The subadditive ergodic theorem and generic stretching factors for free group automorphisms. Preprint, arXiv:math.GR/0504105, to appear in Israel J. Math., 2005. · Zbl 1173.20031 [9] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023 [10] R. Martin. Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups. Ph.D. thesis, University of California, Los Angeles, 1995. [11] J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. (2) 37 (1936), no. 4, 782 – 800. · Zbl 0015.24804
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