A “Tits-alternative” for subgroups of surface mapping class groups

It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket bold upper B bold upper L bold upper M comma bold upper H right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">B</mml:mi> <mml:mi mathvariant="bold">L</mml:mi> <mml:mi mathvariant="bold">M</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\mathbf {BLM},\,\mathbf {H}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). In this paper, the known list of such properties is extended to the “Tits-Alternative”, a property of linear groups established by J. Tits <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket bold upper T right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\mathbf {T}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group. In addition, in order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston’s projective lamination spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket bold upper T bold h Baseline bold 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">T</mml:mi> <mml:mi mathvariant="bold">h</mml:mi> <mml:mn mathvariant="bold">1</mml:mn> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\mathbf {Th1}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This theory generalizes results known for pseudo-Anosov mapping classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket bold upper F bold upper L bold upper P right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> <mml:mi mathvariant="bold">L</mml:mi> <mml:mi mathvariant="bold">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\mathbf {FLP}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.