Abstract.
Let denote the free group of rank 2g. An automorphism φ? Aut(F 2 g ) is generating if N a φ (N b ) = F 2 g , where N a is the normal closure of and N b is defined analogously. We present a characterization of generating automorphisms in Aut(F 2) and observe that there exists a unique (up to equivalence) epimorphism F 2→Z×Z: this is a particular case of the Grigorchuk–Kurchanov conjecture.
This leads to further investigations for splitting homomorphisms for the pairs (F 2 g , F g) and (G g, F g) where G g denotes the fundamental group of a closed orientable surface of genus g and a reformulation of the Poincaré and Grigorchuk–Kurchanov conjectures is derived.
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Received: 1 October 2001
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Ceccherini-Silberstein, T. On the Grigorchuk–Kurchanov conjecture. Manuscripta Math. 107, 451–461 (2002). https://doi.org/10.1007/s002290200246
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DOI: https://doi.org/10.1007/s002290200246