Twist groups of compact 3-manifolds
Among the homeomorphisms of a compact 3-manifold M which act trivially on the homotopy, a typical example is provided by Dehn twists along discs \(D\subset M\) with \(\partial D\subset \partial M\). The twist group of M consists of the isotopy classes of those homeomorphisms of M that are products of such Dehn twists along discs. In many cases, e.g. when M is \({\mathbb{P}}^ 2\)-irreducible, and has non-empty boundary, this twist group is known to be equal to the group of isotopy classes of all homeomorphisms acting trivially on the homotopy. The author completely solves the problem of whether this twist group is finitely generated. More precisely, he shows that the twist group of M is finitely generated if and only if M satisfies the following condition: Each component of the boundary \(\partial M\) contains at most one simple curve (up to isotopy) which bounds a disc in M but does not bound a disc or Möbius strip in \(\partial M\). He also analyses which finitely generated groups can occur as twist groups of 3-manifolds.
finitely generated groups as twist groups of 3-manifolds
scientific article; zbMATH DE number 3927919