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Clusters, currents, and Whitehead’s algorithm. (English) Zbl 1158.20014
The author deals with the automorphism problem of a free group \(F\), i.e., given \(u,v\in F\) whether there exists a \(\tau\in\operatorname{Aut}(F)\) such that \(\tau(u)=v\). In [Ann. Math. (2) 37, 782-800 (1936; Zbl 0015.24804 and JFM 62.1093.04)] J. H. C. Whitehead provided an algorithm that solved this problem: He introduced the so called Whitehead automorphisms by the use of which he proved that two elements \(u,v\in F\) are equivalent under \(\operatorname{Aut}(F)\) (i.e., there exists \(\tau\in\operatorname{Aut}(F)\) such that \(\tau(u)=v\)), if there exists a chain of Whitehead automorphisms which take \(u\) to \(v\). Here the author uses geodesic currents in his study. It is needless to include all the definitions of all used terms and properties in this review.
We shall simply quote his abstract: Using geodesic currents, we provide a theoretical justification for some of the experimental results obtained by Haralick, Miasnikov, and Myasnikov via pattern-recognition methods regarding the behavior of Whitehead’s algorithm on nonminimal inputs, in particular, we prove that the images of “random” elements of a free group \(F\) under the automorphisms of \(F\) form “clusters” that share similar normalized Whitehead graphs and similar behavior with respect to Whitehead’s algorithm.

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