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Fixed $n+1$ to be $n$.
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A. Pascal
A. Pascal
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It is well-known that the braid group $B_{n}$ injects into the group of automorphisms of the free group $F_{n+1}$$F_{n}$. However, there is certainly a kernel when mapping to the outer automorphism group $Out(F_{n+1})$$Out(F_{n})$. Namely, the kernel contains the generator of the center of $B_{n+1}$$B_{n}$. Could someone please explain or give a reference to the fact (?) that the whole kernel of $B_{n} \rightarrow Out(F_{n+1})$$B_{n} \rightarrow Out(F_{n})$ is the center of $B_{n}$?

It is well-known that the braid group $B_{n}$ injects into the group of automorphisms of the free group $F_{n+1}$. However, there is certainly a kernel when mapping to the outer automorphism group $Out(F_{n+1})$. Namely, the kernel contains the generator of the center of $B_{n+1}$. Could someone please explain or give a reference to the fact (?) that the whole kernel of $B_{n} \rightarrow Out(F_{n+1})$ is the center of $B_{n}$?

It is well-known that the braid group $B_{n}$ injects into the group of automorphisms of the free group $F_{n}$. However, there is certainly a kernel when mapping to the outer automorphism group $Out(F_{n})$. Namely, the kernel contains the generator of the center of $B_{n}$. Could someone please explain or give a reference to the fact (?) that the whole kernel of $B_{n} \rightarrow Out(F_{n})$ is the center of $B_{n}$?

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A. Pascal
A. Pascal
  • 1.3k
  • 1
  • 10
  • 20

Center of the braid group and outer automorphisms of the free group

It is well-known that the braid group $B_{n}$ injects into the group of automorphisms of the free group $F_{n+1}$. However, there is certainly a kernel when mapping to the outer automorphism group $Out(F_{n+1})$. Namely, the kernel contains the generator of the center of $B_{n+1}$. Could someone please explain or give a reference to the fact (?) that the whole kernel of $B_{n} \rightarrow Out(F_{n+1})$ is the center of $B_{n}$?