Krull dimension for limit groups. (Q422820)

In this long paper the author develops a dimension theory for varieties defined over free groups. The Krull dimension of a variety \(V\), defined over a nonabelian free group, is the supremum of lengths of chains of irreducible subvarieties contained in \(V\).NEWLINENEWLINE The main result of the paper is: Theorem. There is a function \(D(N)\) such that if NEWLINE\[NEWLINEF_N\twoheadrightarrow L_1\twoheadrightarrow L_2\twoheadrightarrow\cdots\twoheadrightarrow L_kNEWLINE\]NEWLINE is a sequence of proper epimorphisms of limit groups, then \(k\leq D(N)\).NEWLINENEWLINE Here it is not possible to go into details referring all the used terminology and describe the sequence of the arguments. Therefore we make a general comment. To reach this result the author makes a deep study and analysis of previous works on limit groups, especially the work of \textit{Z. Sela} [Publ. Math., Inst. Hautes Étud. Sci. 93, 31-105 (2001; Zbl 1018.20034)]. The main used tool is the theory of JSJ decompositions and here much of the effort is devoted to adjust this theory to the needs of the proof of the above theorem.