Every non-trivial knot group is fully residually perfect

Abstract

Abstract Given a class script upper P <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"> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">P</mml:mi> </mml:mrow> </mml:math> $\mathcal{P}$ of groups we say that a group G is fully residually script upper P <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"> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">P</mml:mi> </mml:mrow> </mml:math> $\mathcal{P}$ if for any finite subset F of G , there exists an epimorphism from G to a group in script upper P <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"> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">P</mml:mi> </mml:mrow> </mml:math> $\mathcal{P}$ which is injective on F . It is known that any non-trivial knot group is fully residually finite, but not residually free. For hyperbolic knots, its knot group is fully residually closed hyperbolic 3–manifold group, and such a group is fully residually simple. In this paper, we show that every non-trivial knot group is fully residually perfect, closed 3–manifold group.