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# Whitehead Problems for Words in $F_n \times \ZZ^m$
We generically call Whitehead problems for a finitely presented group G the problems consisting in, given two objects (of the same certain suitable kind TeXmathcal\O\ ) in G, and a family TeXmathcal\F\ of transformations, decide whether there exists an element in TeXmathcal\F\ sending one object to the other. Specifically we will write TeXtext\WhP$$mathcal\O\,mathcal\F$$ to mean the Whitehead problem with objects in TeXmathcal\O\ and transformations in TeXmathcal\F\ , i.e., TeXdisplaystyle\text\WhP$$mathcal\O\,mathcal\F$$ equiv text\textquestiondown \exists varphi in mathcal\F\text\ such that \o\_\1\mathop\longmapsto \limitsˆ\varphi \o\_\2\text\?\,\_\ (o\_\1\,o\_\2\text\ in \mathcal\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.The kind of “objects in G” usually considered includes elements (i.e., words in the generators), subgroups, and conjugacy clas