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Register a new userCentralizers in Mapping Class Group and decidability of Thurston Equivalence
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We find a constructive bound for the word length of a generating set for the centralizer of an element of the Mapping Class Group. As a consequence, we show that it is algorithmically decidable whether two postcritically finite branched coverings of the sphere are Thurston equivalent.
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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 $text{Mod}(S_g)$.
These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers The second rational homology group of the moduli space of curves with level structures and The Picard group of the moduli space of curves with level structures.
We prove that various subgroups of the mapping class group $Mod(Sigma)$ of a surface $Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the point-pushing and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Sigma;mathbb{Z})$.
We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g times 2g$ symplectic group for $L$ odd and $g geq 3$.
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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