Mathematical Research Letters
Volume 13 (2006)
Countable groups are mapping class groups of hyperbolic $3$-manifolds
Pages: 897 – 910
We prove that for every countable group $G$ there exists a hyperbolic $3$-manifold $M$ such that the isometry group of $M$, the mapping class group of $M$, and the outer automorphism group of $\pi_1 (M)$ are isomorphic to $G$.