Bounded cohomology and scl of verbal wreath products

Abstract Given a countable group A with trivial bounded cohomology for all semi-separable coefficients and any group normal upper Gamma <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:mi mathvariant="normal">Γ</mml:mi> </mml:math> $\Gamma$ , we prove that the bounded cohomology of certain restricted verbal wreath products normal upper Gamma wreath product Superscript Super Subscript upper W Baseline upper A <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:mi mathvariant="normal">Γ</mml:mi> <mml:msup> <mml:mo>≀</mml:mo> <mml:mrow> <mml:msub> <mml:mi/> <mml:mi>W</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mi>A</mml:mi> </mml:math> $\Gamma \wr^{_W} A$ vanishes in positive degrees, for all semi-separable coefficients. Examples of such verbal wreath products include the standard restricted wreath product (extending a recent result by Monod for lamplighters groups) as well as verbal wreath products arising from n -solvable, n -nilpotent and k -Burnside ( k equals 2 comma 3 comma 4 comma 6 <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:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> </mml:math> $k = 2,3,4,6$ ) verbal products. Moreover, we prove that the stable commutator length of any verbal wreath product normal upper Gamma wreath product Superscript Super Subscript upper W Baseline upper A <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:mi mathvariant="normal">Γ</mml:mi> <mml:msup> <mml:mo>≀</mml:mo> <mml:mrow> <mml:msub> <mml:mi/> <mml:mi>W</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mi>A</mml:mi> </mml:math> $\Gamma \wr^{_W} A$ vanishes. As an application, we show that every group of type upper F Subscript 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:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> $F_p$ isometrically embeds into a group of type upper F Subscript 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:msub> <mml:mi>F</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> $F_p$ with vanishing bounded cohomology in degree n greater than or equals 1 <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:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> $n\geq 1$</jats:tex-ma