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Register a new userOn finite index subgroups of a universal group
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The orbifold group of the Borromean rings with singular angle 90 degrees, $U$, is a universal group, because every closed oriented 3--manifold $M^{3}$ occurs as a quotient space $M^{3} = H^{3}/G$, where $G$ is a finite index subgroup of $U$. Therefore, an interesting, but quite difficult problem, is to classify the finite index subgroups of the universal group $U$. One of the purposes of this paper is to begin this classification. In particular we analyze the classification of the finite index subgroups of $U$ that are generated by rotations.
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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.
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.
Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $delta_k$-values $a,bin G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $Pcap G^{(k)}$ is generated by $delta_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.
Let $S$ be a compact orientable surface, and $Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b in Mod(S)$ are independent (i.e., $[a^n,b^m] ot=1$ for any $n,m ot=0$) pseudo-Anosov elements. Then for any $n,m ge M$, the subgroup $<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $<a^n,b^m>$ is free of rank two and convex-cocompact if $|n|+|m| ge N$ and $nm ot=0$.
In this note we make progress toward a conjecture of Durham--Fanoni--Vlamis, showing that every infinite-type surface with finite-invariance index 1 and no nondisplaceable compact subsurfaces fails to have a good curve graph, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and where the natural mapping class group action has infinite diameter orbits. Our arguments use tools developed by Mann--Rafi in their study of the coarse geometry of big mapping class groups.
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