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The Bieri-Neumann-Strebel invariant for basis-conjugating automorphisms of free groups. (English) Zbl 0963.20016

Let \(F_n\) be a free group of rank \(n\) with the set \(X_n=\{x_1,\dots,x_n\}\) of free generators. The group \(P\Sigma_n\), a subgroup of \(\operatorname{Aut} F_n\), consists of all \(\alpha\in\operatorname{Aut} F_n\) such that \(\alpha(x_i)\) is conjugated to \(x_i\), for \(i=1,\dots,n\). J. McCool derived [in Can. J. Math. 38, 1525-1529 (1986; Zbl 0613.20024)] a finite presentation for \(P\Sigma_n\), and M. Gutiérrez and S. Krstić found [in Int. J. Algebra Comput. 8, No. 6, 631-669 (1998; Zbl 0960.20021)] normal forms for the elements of \(P\Sigma_n\).
In this paper, the author uses actions on \(\mathbb{R}\)-trees to determine the Bieri-Neumann-Strebel invariant of \(P\Sigma_n\). The BNS-invariant tells, among other things, which subgroups of \(P\Sigma_n\) with Abelian quotient are finitely generated.

MSC:

20F28 Automorphism groups of groups
20E05 Free nonabelian groups
20E08 Groups acting on trees
20F05 Generators, relations, and presentations of groups
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