an:06283581
Zbl 1319.20030
Kapovich, Ilya
Algorithmic detectability of iwip automorphisms.
EN
Bull. Lond. Math. Soc. 46, No. 2, 279-290 (2014).
0024-6093 1469-2120
2014
j
20F10 20E36 20E05
algorithms; finitely generated free groups; iwip automorphisms; fully irreducible automorphisms
From the introduction: The notion of a pseudo-Anosov homeomorphism of a compact surface plays a fundamental role in low-dimensional topology and the study of mapping class groups. In the context of \(\mathrm{Out}(F_N)\), the concept of being pseudo-Anosov has several (nonequivalent) analogs.
The first is the notion of an `atoroidal' automorphism. An element \(\varphi\in\mathrm{Out}(F_N)\) is called atoroidal if there do not exist \(m\geq 1\), \(h\in F_N\), \(h\neq 1\) such that \(\varphi^m\) preserves the conjugacy class \([h]\) of \(h\) in \(F_N\). Another, more important, free group analog of being pseudo-Anosov is the notion of a `fully irreducible' or `iwip' automorphism. An element \(\varphi\in\mathrm{Out}(F_N)\) is called reducible if there exists a free product decomposition \(F_N=A_1*\cdots*A_k*C\) with \(k\geq 1\), \(A_i\neq 1\) and \(A_i\neq F_N\) such that \(\varphi\) permutes the conjugacy classes \([A_1],\ldots,[A_k]\). An element \(\varphi\in\mathrm{Out}(F_N)\) is irreducible if it is not reducible. An element \(\varphi\in\mathrm{Out}(F_N)\) is fully irreducible or iwip (which stands for `irreducible with irreducible powers') if \(\varphi^m\) is irreducible for all integers \(m\geq 1\) (equivalently, for all nonzero integers \(m\)). Thus, \(\varphi\) is an iwip if and only if there do not exist a proper free factor \(A\) of \(F_N\) and \(m\geq 1\) such that \(\varphi^m([A])=[A]\).
There is no obvious approach to algorithmically deciding whether an element\(\varphi\in\mathrm{Out}(F_N)\) is an iwip. In this note, we provide such an algorithm.
Theorem A. There exists an algorithm that, given \(N\geq 2\) and \(\varphi\in\mathrm{Out}(F_N)\), decides whether or not \(\varphi\) is an iwip.