Two-Dimensional Folding for Graphs of Free Groups
نویسنده
چکیده
Several classical facts due to Lyndon, Shenitzer, and Baumslag about ranks of subgroups generated by solutions to equations over the free group are generalized and given a uniform proof. Given a graph of free groups G = Γ(Fi) over cyclic subgroups and a homomorphism φ : G → F which embeds each Fi, there is a natural two complex Xφ which encodes φ. Our strategy for extracting algebraic information about G is to find a representative φ of φ in Aut(F)◦φ such that Xφ′ is as simple as possible. This is accomplished by translating the effect of an automorphism of F into a move called folding, in honor of Stallings’ folding, on the graph of spaces Xφ. This is important for forthcoming work on the Krull dimension of limit groups.
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