Abstract
<p>In this two part work we prove that for every finitely generated subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma greater-than sans-serif upper O sans-serif u sans-serif t left-parenthesis upper F Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">O</mml:mi> <mml:mi mathvariant="sans-serif">u</mml:mi> <mml:mi mathvariant="sans-serif">t</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma >{\mathsf {Out}}(F_n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is virtually abelian or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript b Superscript 2 Baseline left-parenthesis normal upper Gamma semicolon double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^2_b(\Gamma ;{\mathbb {R}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a vector space embedding of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The method uses actions on hyperbolic spaces. In Part I we focus on the case of infinite lamination subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>—those for which the set of all attracting laminations of all elements of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an infinite set—using actions on free splitting complexes of free groups. In Part II we focus on finite lamination subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and on the construction of useful new hyperbolic actions of those subgroups.</p>