Algorithmic detectability of iwip automorphisms. (English) Zbl 1319.20030

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.


20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
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