Polynomial growth and subgroups of \(\mathrm{Out}(F_N)\) (Q6047214)

Let \(F_{n}\) be the free group on \(n\) generators and let \(\mathrm{Out}(F_{n})\) be the outer automorphism group of \(F_{n}\). The paper under review, which is a continuation of [Conform. Geom. Dyn. 27, 161--263 (2023; Zbl 1523.20038)] by the same author, is dedicated to the study of the exponential growth of elements in \(\mathrm{Out}(F_{n})\). An element \(\phi \in \mathrm{Out}(F_{n})\) is exponentially growing if there exist a conjugacy class \([g] \subseteq F_{n}\), a free basis \(\mathfrak{B}\) of \(F_{n}\) and a constant \(K > 0\) such that, for every \(m \in \mathbb{N}^{\ast}\) the inequality \(\ell_{\mathfrak{B}}(\phi^{m}([g]) \geq e^{Km}\) is satisfied, where \(\ell_{\mathfrak{B}}(\phi^{m}([g])\) denotes the length of a cyclically reduced representative of \(\phi^{m}([g])\) in the basis \(\mathfrak{B}\). An element \(g \in F_{n}\) is exponentially growing under iteration of \(\phi\) if \(\ell_{\mathfrak{B}}(\phi^{m}([g]) \geq e^{Km}\). Otherwise, one can show, using the train track method [\textit{M. Bestvina} and \textit{M. Handel}; Ann. Math. (2) 135, No. 1, 1--51 (1992; Zbl 0757.57004)], that \(\ell_{\mathfrak{B}}(\phi^{m}([g]) \leq (m+1)^{K}\) and in this case, \(g\) is said to have polynomial growth under iteration of \(\phi\). Let \(\mathrm{Poly}(\phi)\) be the set of elements of \(F_{n}\) which have polynomial growth under iteration of \(\phi\) and, if \(H \leq \mathrm{Out}(F_{n})\), let \(\mathrm{Poly}(H)=\bigcap_{\phi \in H} \mathrm{Poly}(\phi)\). The main result of this paper is Theorem 1.1: If \(n \geq 2\) and \(H \leq \mathrm{Out}(F_{n})\), then there is an element \(\phi \in H\) such that \(\mathrm{Poly}(\phi) = \mathrm{Poly}(H)\).