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The frequency space of a free group. (English) Zbl 1110.20031
Let \(F(X)\) be a free group of rank \(k\geq 2\) with a fixed free basis \(X\). For any \(m\geq 0\) let \(D(m)\) denote the number of elements of length \(m\) (with respect to the basis \(X\)) in \(F\). For \(m\geq 1\) let \(Q(m)\) be a compact convex polyhedron in \(\mathbb{R}^{D(m)}\) (see Definition 2.4 in the paper). For \(m\geq 1\) it are defined maps \(\pi_m\colon Q(m)\to Q(m-1)\). These maps are defined in an “appropriate” way, so there exists the inverse limit of the sequence \[ \cdots\to Q(m)\to Q(m-1)\to\cdots \to Q(1)\to Q(0):=\{1\} \] and it is defined to be the ‘frequency space’ \[ Q(F):=\varprojlim(Q_m,\pi_m). \] A ‘cyclic word’ in \(F\) represents all the words in \(F\) which are obtained by cyclic permutations of the letters of a cyclically reduced word in \(F\). For a non trivial cyclic word \(w\) and for a cyclically reduced word \(u\) it is defined the ‘frequency’ \(f_w(u)\) of \(u\) in \(w\). Let \(C\) be the set of all cyclic words in \(F\), then for \(m\geq 1\) are defined maps \(\alpha_m\colon C\to Q_m\) (Definition 2.8). These maps define a map \(\alpha\colon C\to Q(F)\). A systematic study of the maps \(\alpha_m\) and \(\alpha\) is given in Paragraph 4 of the paper with the consideration of ‘initial graphs’ for elements in \(Q_m\), \(m\geq 1\) (Definition 4.1). There are described the elements of \(Q_m\) which are images of cyclic words in \(F\) under the map \(\alpha_m\) (Theorem 4.1). It is proved that the maps \(\pi_m\colon Q_m\to Q_{m-1}\) are onto, the set \(\alpha(C)\) is dense in \(Q(F)\) and that the external elements of the polyhedron \(Q_m\) belong to \(\alpha_m(C)\).
For the free group \(F(X)\) let \(\partial F\) denote the hyperbolic boundary of \(F\), which is identified with the set of semi-infinite freely reduced words in the base \(X^{\pm 1}\). If \(T\colon\partial F\to\partial F\) is the map which erases the first letter of every semi-infinity word, then it is proved that the frequency space \(Q(F)\) is canonically identified with the space of all \(T\)-invariant Borel probability measures on \(\partial F\) (Theorem 3.2).
In Theorem 5.6 it is proved that the group \(\text{Out}(F)\) of outer automorphisms of \(F\) acts by homeomorphisms on \(Q(F)\). The arguments developed to prove this theorem enable the author to prove the following theorems.
Theorem 6.3. For any \(\varphi\in\text{Out}(F)\) and \(1\neq w\in F\) there exist positive rational numbers \(\lambda,\mu\), algorithmically computable in terms of \(\varphi\), such that \(\lambda\leq\|\varphi(w)\|/\| w\|\leq\mu\), where \(\|\cdot\|\) denotes the word length for a given basis. Moreover, for every rational number \(\lambda\leq r\leq\mu\) there exists a nontrivial cyclic word \(w\) with \(\|\varphi(w)\|/\| w\|=r\).
Theorem 6.2. For an outer automorphism \(\varphi\) let \[ \lambda_0(\varphi)=\inf\{\max(\|\varphi(w)\|/\| w\|,\|\varphi^{-1}(w)\|/\| w\|)\mid 1\neq w\in F\}. \] Then \(\lambda_0(\varphi)\) is a rational number, algorithmically computable in terms of \(\varphi\). The outer automorphism \(\varphi\) is strictly hyperbolic if and only if \(\lambda_0(\varphi)>1\). (An outer automorphism \(\varphi\) is ‘strictly hyperbolic’ if there exist \(\lambda>1\) such that \(\lambda\| w\|\leq\max\{\|\varphi(w)\|, \|\varphi^{-1}(w)\|\}\) for every cyclic word \(w\in F\).)
The paper concludes with a brief discussion on geodesic currents on free groups and it is proved that the frequency space \(Q_X(F)\) is equivariantly homeomorphic to the space \(\text{Curr}(F)\) of geodesic currents (Lemma 7.2). This homeomorphism depends on the choice of the basis \(X\) of the free group. An action of \(\operatorname{Aut}(F)\) is defined on \(\text{Curr}(F)\), this action factors through an action of \(\text{Out}(F)\) on \(\text{Curr}(F)\). For a fixed basis \(X\) of \(F\) this action is the same (via the homeomorphism of Lemma 7.2) as the action of \(\text{Out}(F)\) on \(Q_X(F)\) defined above.
R. Martin [“Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups”, PhD Thesis (1995)] studied the space of geodesic currents on a free group \(F\) and he obtained an action of \(\text{Out}(F)\) on \(\text{Curr}(F)\), which coincides with the action constructed in this paper. A more systematic study of the connection of the frequency space \(Q_X(F)\) and the geodesic currents \(\text{Curr}(F)\) has been done by the author in a recent paper [in Contemp. Math. 394, 149-176 (2006; see the review Zbl 1110.20034 below)].

20F65 Geometric group theory
20E05 Free nonabelian groups
20F28 Automorphism groups of groups
20E36 Automorphisms of infinite groups
20F05 Generators, relations, and presentations of groups
57M05 Fundamental group, presentations, free differential calculus
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