Commensurations of \(\mathrm{Aut}(F_N)\) and its Torelli subgroup (Q6614090)

A commensuration of a group \(G\) is an isomorphism between two finite-index subgroups of \(G\) and two commensurations are equivalent if they agree on a common finite-index subgroup of their domains. The set of all commensurations with this equivalence relation forms a group \(\mathrm{Comm}(G)\) called the abstract commensurator of \(G\). There is a natural homomorphism \(\mathrm{ad}: G \rightarrow \mathrm{Comm}(G)\) induced by conjugation, if \(\mathrm{ad}\) is an epimorphism, then \(G\) is said to be commensurator-rigid.\N\NLet \(F_{N}\) be the free group of (finite) rank \(N\); \textit{B. Farb} and \textit{M. Handel}, in [Publ. Math., Inst. Hautes Étud. Sci. 105, 1--48 (2007; Zbl 1137.20018)] (for \(N\geq 4\)), and \textit{C. Horbez} and \textit{R. D. Wade}, in [Trans. Am. Math. Soc. 373, No. 4, 2699--2742 (2020; Zbl 1452.20030)] (for \(N \geq 3\)) proved that \(\mathrm{Out}(F_{N})\), the outer automorphism group of \(F_{N}\), is commensurator-rigid.\N\NThe main purpose of the paper under review is to prove that \(\mathrm{Aut}(F_{N})\) is also commensurator rigid for \(N \geq 3\). In fact, the authors prove Theorem A: For all \(N \geq 3\), the natural map \(\mathrm{ad}: \mathrm{Aut}(F_{N})\rightarrow \mathrm{Comm}(\mathrm{Aut}(F_{N}))\) is an isomorphism. More precisely, any isomorphism \(f : \Gamma_{1} \rightarrow \Gamma_{2}\) between finite-index subgroups of \(\mathrm{Aut}(F_{N})\) is the restriction of conjugation by an element of \(\mathrm{Aut}(F_{N})\).\N\NA subgroup \(\Gamma < \mathrm{Aut}(F_{N})\) is ample if \(\Gamma\) contains a non-trivial power of every Nielsen automorphism (hence a non-trivial power of the inner automorphism determined by each primitive element). In particular all finite-index subgroups of \(\mathrm{Aut}(F_{N})\) are ample. The relative commensurator of a subgroup \(\Gamma < G\) is the group \(\mathrm{Comm}_{G}(\Gamma)=\big \{ g \in G \; \big | \; |\Gamma:\Gamma \cap \Gamma^{g}| < \infty, \, |\Gamma^{g} :\Gamma \cap \Gamma^{g}| < \infty \big \}\).\N\NThe second main result of this paper is Theorem B: If \(\Gamma\) is an ample subgroup of \(\mathrm{Aut}(F_{N})\) then the natural map \(\mathrm{Comm}_{\mathrm{Aut}(F_{n})}(\Gamma) \rightarrow \mathrm{Comm}(\Gamma)\) is an isomorphism. Furthermore, any isomorphism \(f : \Gamma_{1} \rightarrow \Gamma_{2}\) between finite-index subgroups of \(\Gamma\) is the restriction of conjugation by an element of \(\mathrm{Aut}(F_{N})\).\N\NThe Torelli subgroup of \(\mathrm{IA}_{N}\) of \(\mathrm{Aut}(F_{N})\) is the subgroup acting trivially on \(H_{1}(F_{N}) = F_{N}/[F_{N},F_{N}]\). The group \(\mathrm{IA}_{N}\) is not large, but the authors are able to prove Theorem D: For all \(N \geq 3\), the natural map \(\mathrm{Aut}(F_{N}) \rightarrow \mathrm{Comm}(\mathrm{IA}_{N})\) is an isomorphism. Furthermore, any isomorphism between finite-index subgroups of \(\mathrm{IA}_{N}\) is the restriction of conjugation by an element of \(\mathrm{Aut}(F_{N})\).\N\NFinally, the authors provide a geometric proof of the result of \textit{J. L. Dyer} et al. [Arch. Math. 38, 404--409 (1982; Zbl 0476.20025)], thus obtaining Theorem E: The abstract commensurator of \(\mathrm{Aut}(F_{2})\) is isomorphic to the extended mapping class group of a sphere with five punctures. Furthermore, \(\mathrm{Aut}(F_{2})\) is an index-five subgroup of its abstract commensurator.