Countable groups are mapping class groups of hyperbolic 3-manifolds (Q862065)

For a hyperbolic manifold \(M\) denote by \(\text{Isom}(M)\) its group of isometries, by \(\mathcal{MCG}(M)\) its mapping class group, that is the group of isotopy classes of self-homeomorphisms of \(M\), and by \(\text{Out}(\pi_{1}(M))\) the outer automorphism group of its fundamental group. The main result in the paper under review says that for every countable group \(G\) there exists a hyperbolic \(3\)-manifold \(M\) such that \(G \cong \text{Isom}(M) \cong \mathcal{MCG}(M) \cong \text{Out}(\pi_{1}(M))\). An interesting corollary of this theorem is that there exist uncountably many non-isomorphic groups that are fundamental groups of hyperbolic \(3\)-manifolds. The main theorem is proved by constructing for each countable group \(G\) a decorated \(2\)-dimensional polyhedron whose combinatorial automorphism group is the countable group \(G\) in question. Then a hyperbolic \(3\)-manifold \(M\) is associated with each decorated polyhedron such that the isometry group of the manifold is isomorphic to the combinatorial automorphism group of the polyhedron (and therefore is isomorphic to \(G\)). Now the hyperbolic manifolds arising from this particular construction are shown to satisfy certain rigidity criteria which allow the authors to prove that the isometry group of \(M\) is isomorphic to both its mapping class group and its outer automorphism group. The main theorem was previously proved for finite groups by \textit{S. Kojima} [Topology Appl. 29, No. 3, 297--307 (1988; Zbl 0654.57006)], who proved that every finite group is the isometry group of a compact hyperbolic \(3\)-manifold. Using the rigidity results by \textit{G. D.~Mostow} [Publ. Math., Inst. Hautes Étud. Sci. 34, 53--104 (1968; Zbl 0189.09402)] and results by \textit{D.~Gabai, G. R.~Meyerhoff} and \textit{N.~Thurston} [Ann. Math. (2) 157, No. 2, 335--431 (2003; Zbl 1052.57019)] one can further conclude that the outer automorphism group, the isometry group and the mapping class group are all isomorphic finite groups under these assumptions.