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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 \(L(\varphi_n)\to\infty\).

20F65 Geometric group theory
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
Full Text: DOI arXiv
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