On the Grigorchuk–Kurchanov conjecture
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Let \(\) denote the free group of rank 2g. An automorphism φ? Aut(F2g) is generating if Naφ (Nb) = F2g, where Na is the normal closure of \(\) and Nb is defined analogously. We present a characterization of generating automorphisms in Aut(F2) and observe that there exists a unique (up to equivalence) epimorphism F2→Z×Z: this is a particular case of the Grigorchuk–Kurchanov conjecture.
This leads to further investigations for splitting homomorphisms for the pairs (F2g, Fg) and (Gg, Fg) where Gg 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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