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O1 ;o2 in O/ : It is customary to include as a part of the problem the search of one of such transformations, in case that it exists. So will we. , words in the generators), subgroups, and conjugacy classes, as well as tuples of these, while the typical families of transformations (which we will always think of acting on the right) are those of automorphisms, monomorphisms, epimorphisms and endomorphisms of G. We denote them respectively by Aut G, Mon G, Epi G and End G. Fn ; Aut Fn /, where Fn denotes the free group on n generators.

Math. J. 60, 1905–1926 (2011) 5. A. Fossas, M. Nguyen, Thompson’s group T is the orientation-preserving automorphism group of a cellular complex. Publ. Math. 56(2), 305–326 (2012) 6. L. Funar, C. Kapoudjian, V. Sergiescu, Asymptotically rigid mapping class groups and Thompson’s groups, in Handbook of Teichmüller Theory, vol. IV, ed. by A. Papadopoulos (European Mathematical Society Publishing House, Boston, 2012), pp. 595–664 7. E. Ghys, V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle.

Theorem 2. Fn ; End Fn / is solvable. t u So, the auto, mono and endo Whitehead problems (for words) are solvable for both Zm and Fn . For G D Zm Fn though, these problems turn out to be more than the mere juxtaposition of the corresponding problems for its factors. This is because the endomorphisms of G are more than pairs of endomorphisms of Zm and Fn as well. It is not difficult to obtain a complete description of them imposing preservation of the (commutativity) relations defining G. Proposition 3.

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