Assume $V$ be a genus larger than 1 handlebody, $S=\partial_{+} V$.
Denote $N$ be the normal closure of $MCG(V)$ in $MCG(S)$.
Is there any material related to the quotient group $MCG(S)/N$ ? Thanks!
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The normal closure of $MCG(V)$ in $MCG(S)$ is all of $MCG(S)$. To see why, we know that $MCG(S)$ is generated by Dehn twists, so it suffices to prove that each Dehn twist is in the normal closure. Consider the Dehn twist $\tau_c$ about an essential simple closed curve $c$. There exists a homeomorphism $f : S \to S$ such that $c' = f(c)$ bounds a properly embedded disc $D \subset V$, and $\tau_{c'}$ extends to a twist about the disc $D$, so $\tau_c = f^{-1} \circ \tau_{c'} \circ f$ is in the normal closure of $MCG(V)$. To find this homeomorphism, observe that the there is one $MCG(S)$-orbit of nonseparating curves, and for each partition $genus(S)=m+n$ there is one $MCG(S)$ orbit of separating curves whose complements have genus $m,n$. Each such orbit clearly has a representative bounding a disc in $V$. |
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