Alternating and Symmetric Separability of Free Products

Let F ∗ G F G upper F asterisk upper G be a free product of a free group F and a LERF group G . We provide sufficient conditions for a subgroup H of F ∗ G F G upper F asterisk upper G to be A ∪ S A∪ S script upper A union script upper S -separable, that is, for any finite set { γ 1 , … , γ n } ⊂ ( F ∗ G ) ∖ H \γ ₁, … , γ ₙ\ ⊂ (F G) H StartSet gamma 1 comma ellipsis comma gamma Subscript n Baseline EndSet subset of left parenthesis upper F asterisk upper G right parenthesis minus upper H , there is a surjection f from F ∗ G F G upper F asterisk upper G to an alternating or symmetric group such that f ( γ i ) ∉ f ( H ) f(γ ᵢ) ∉ f(H) f left parenthesis gamma Subscript i Baseline right parenthesis not an element of f left parenthesis upper H right parenthesis for all i . As a corollary, any finitely generated infinite-index subgroup of a free group is A