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Distortion and entropy for automorphisms of free groups. (English) Zbl 1220.37018
From the author’s abstract: Recently, several numerical invariants have been introduced to characterize the distortion induced by automorphisms of a free group. We unify these by interpreting them in terms of an entropy function of a kind familiar in thermodynamic ergodic theory. We draw an analogy between this approach and the Manhattan curve associated to a pair of hyperbolic surfaces.
Let \(F\) be a free group of rank \(n\) on \(n\) free generators \(\{a_1,\dots,a_n\}=\mathcal A\). If \(\varphi\) is an automorphism of \(F\), then certain invariants in terms of \(\varphi\) are introduced. Let \(|x|\) be the usual length of \(x\in F\) in terms of \(a_1,\dots,a_n\) and \(\|w\|=\min\{|x|:x\in w\}\), \(w\in \mathcal C(F)\), \(\mathcal C(F)\) being the set of nontrivial conjugacy classes in \(F\). Let \(\mathcal D_\varphi=\{\frac{\|\varphi (w)\|}{\|w\|}:w\in\mathcal C(F)\}\). One of the main results is in terms of the Manhattan curve \(\mathfrak M_\varphi\) which is the boundary of the set \[ \{(a,b)\in \mathbb R^2:\sum_{w\in\mathcal C(F)}e^{-a|w|-b|\varphi(w)|}<+\infty\}. \] Thus this theorem goes as follows:
(i) \(\mathfrak M_\varphi\) is a straight line if and only if \(\varphi\) is simple.
(ii) \(\mathfrak M_\varphi\) is real analytic.
(iii) \(\mathfrak M_\varphi\) has asymptotes whose normals have slopes equal to the \(\max\mathcal D_\varphi\) and \(\min\mathcal D_\varphi\).
(iv) \(\mathfrak M_\varphi\) passes through \((\log(2k-1),0)\), where its normal has slope equal to the generic stretch \(\lambda(\varphi)\).
(v) There is a unique point \((a,b)\in\mathfrak M_\varphi\) where the normal has slope 1 and \(a+b=\mathfrak h(1)\).
Here an automorphism \(\varphi\) is called simple if it is the product of an inner automorphism and a permutation automorphism. A permutation automorphism is one that sends each generator to some generator or the inverse of some generator, \(\lambda(\varphi)=\lim _{n\to+\infty }\frac{|\varphi(x_0x_1\cdots x_{n-1}|}{n}\), \(x_i\in\mathcal A\cap\mathcal A^{-1}\) and \(\mathfrak h\) a strictly concave analytic function \(\mathfrak h:\text{int}(\overline{\mathcal D(\varphi)})\to\mathbb R^+\).

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C35 Orbit growth in dynamical systems
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20F69 Asymptotic properties of groups
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