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Register a new userAll finite groups are involved in the Mapping Class Group
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Let $Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $ggeq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of $Gamma_g$.
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For $ggeq 2$, let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give a theoretical algorithm for computing its roots up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $text{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $text{Mod}(S_g)$, and the commutator subgroup of $text{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, realized by the roots of $T_c^{q(F)}$, where $c$ is a separating curve in $S_g$ of genus $[g/2]$ and $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Finally, for $ggeq 3$ we show that any pseudo-periodic having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $text{Mod}(S_g)$. Consequently, we establish there always exist roots of bounding pair maps and powers of Dehn twists that normally generate $text{Mod}(S_g)$.
We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.
We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.
We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the mapping class group is trivial.
For some $g geq 3$, let $Gamma$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Gamma$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n geq 1$, we have $T_x^n in Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $Gamma$. Our second result shows that the abelianization of $Gamma$ is finite if $Gamma$ contains a large chunk (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. This generalizes work of Hain and Boggi.
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