Outer automorphism groups of free groups: linear and free representations. (Q2840174)

Let \(F_n\) be the free group of rank \(n\). Although for \(n\leq m\) there are natural embeddings \(\Aut(F_n)\hookrightarrow\Aut(F_m)\), it is not obvious if there is an embedding \(\text{Out}(F_n)\hookrightarrow\text{Out}(F_m)\), where \(\text{Out}(F_n)\) is the outer automorphism group of the free group \(F_n\).NEWLINENEWLINE \textit{D. G. Khramtsov} [in Group-theoretic studies. Collection of scientific works. Sverdlovsk: Ural'skoe Otdelenie AN SSSR, 95-127 (1990; Zbl 0808.20035)] proved that there is no embedding from \(\text{Out}(F_n)\) into \(\text{Out}(F_{n+1})\) (\(n>1\)).NEWLINENEWLINE \textit{M. R. Bridson} and \textit{K. Vogtmann}, [in Bull. Lond. Math. Soc. 35, No. 6, 785-792 (2003; Zbl 1049.20021)], proved that, for any \(n\geq 3\) and \(m<n\), there exist no embeddings \(\text{Out}(F_n)\hookrightarrow\text{Out}(F_m)\).NEWLINENEWLINE In [\textit{M. R. Bridson} and \textit{K. Vogtmann}, Math. Ann. 353, No. 4, 1069-1102 (2012; Zbl 1261.20035)] it is proved that the image of any homomorphism \(\text{Out}(F_n)\to\text{Out}(F_m)\) is contained in \(\mathbb Z_2\), the cyclic group of order 2, provided that \(n\geq 9\), \(m\neq n\), and \(m\leq 2n-2\) when \(n\) is odd, or \(m\leq 2n\) when \(n\) is even.NEWLINENEWLINE In this paper the author straights this last result and provesNEWLINENEWLINE Theorem A. Let \(n,m\in\mathbb N\) be distinct, \(n\geq 6\), \(m<{n\choose 2}\), and let \(\varphi\colon\text{Out}(F_n)\to\text{Out}(F_m)\) be a homomorphism. Then the image of \(\varphi\) is contained in a copy of \(\mathbb Z_2\), the finite group of order 2.NEWLINENEWLINE Using a result of \textit{M. R. Bridson} and \textit{B. Farb} [in Topology Appl. 110, No. 1, 21-24 (2001; Zbl 0964.22011)] the author extends the result above and provesNEWLINENEWLINE Theorem B. Let \(n,m\in\mathbb N\) be distinct, \(n\) even and at least 6. Let \(\varphi\colon\text{Out}(F_n)\to\text{Out}(F_m)\) be a homomorphism. Then the image of \(\varphi\) is finite, provided that \({n\choose 2}\leq m<{n+1\choose 2}\).NEWLINENEWLINE To obtain these results, the author investigates the low-dimensional representation theory of \(\text{Out}(F_n)\) and provesNEWLINENEWLINE Theorem C. Let \(\mathbb K\) be a field of characteristic equal to zero or greater than \(n+1\). Suppose \(\varphi\colon\text{Out}(F_n)\to\text{GL}(V)\) is an \(m\)-dimensional \(\mathbb K\)-linear representation of \(\text{Out}(F_n)\), where \(n\geq 6\) and \(m<{n+1\choose 2}\). Then \(\varphi\) factors through the natural projection \(p\colon\text{Out}(F_n)\to\text{GL}_n(\mathbb Z)\).NEWLINENEWLINE After that he deals with allowed representations of a finite subgroup of \(\text{Out}(F_n)\). Using a result of Culler, Khramtsov and Zimmermann (proved independently) he realizes the action of this finite group on the conjugacy classes of \(F_m\) as induced by an action on a finite graph. The comparison of the representation theory with the action on the homology of this graph yields Theorem A.NEWLINENEWLINE As the author points out the question of finding \(n\) for which there exists an embedding \(\text{Out}(F_n)\hookrightarrow\text{Out}(F_{2n})\) has not been fully answered. Also the cases \(n=3,4,5\) remain unanswered, except his partial (unpublished) result that \(\text{Out}(F_3)\) does not embed into \(\text{Out}(F_5)\).