ALTERNATING AND SYMMETRIC SEPARABILITY OF FREE PRODUCTS

Abstract Let <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>F</mml:mi> <mml:mo>∗</mml:mo> <mml:mi>G</mml:mi> </mml:math> $F \ast G$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>F</mml:mi> <mml:mo>∗</mml:mo> <mml:mi>G</mml:mi> </mml:math> $F \ast G$ upper F asterisk upper G to be <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo>∪</mml:mo> <mml:mrow> <mml:mi mathvariant="script">S</mml:mi> </mml:mrow> </mml:math> $\mathcal A\cup \mathcal S$ script upper A union script upper S -separable, that is, for any finite set <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false" fence="false">{</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false" fence="false">}</mml:mo> <mml:mo>⊂</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>∗</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∖</mml:mo> <mml:mi>H</mml:mi> </mml:math> $\{\gamma _1, \ldots , \gamma _n\} \subset (F \ast G) \setminus H$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>F</mml:mi> <mml:mo>∗</mml:mo> <mml:mi>G</mml:mi> </mml:math> $F \ast G$ upper F asterisk upper G to an alternating or symmetric group such that <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∉</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $f(\gamma _i) \notin f(H)$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="script">A</mml:mi> </mml:mrow>