On rank of the join of two subgroups in a free group

2016/08/01 by Ivanov, Sergei V.
#20E05 #20E07 #20F65 #57M07 (Primary) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1608.00617

Abstract

Let H, K be two finitely generated subgroups of a free group, let ⟨ H, K ⟩ denote the subgroup generated by H, K, called the join of H, K, and let neither of H, K have finite index in ⟨ H, K ⟩. We prove the existence of an epimorphism ζ: ⟨ H, K ⟩ → F₂, where F₂ is a free group of rank 2, such that the restriction of ζ on both H and K is injective and the restriction ζ₀ : H ∩ K → ζ(H) ∩ ζ(K) of ζ on H ∩ K to ζ(H) ∩ ζ(K) is surjective. This is obtained as a corollary of an analogous result on rank of the generalized join of two finitely generated subgroups in a free group.

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