Let B be a 3-dimensional handlebody of genus g. Let ℳ be the group of the isotopy classes of orientation preserving homeomorphisms of B. We construct a 2-dimensional simplicial complex X, connected and simply-connected, on which ℳ acts by simplicial transformations and has only a finite number of orbits. From this action we derive an explicit finite presentation of ℳ.
It is known that a finite 2-group acting on a ℤ₂-homology 3-sphere has at most ten conjugacy classes of involutions; the action of groups with the maximal number of conjugacy classes of involutions is strictly related to some questions concerning the representation of hyperbolic 3-manifolds as 2-fold branched coverings of knots. Using a low-dimensional approach we classify these maximal actions both from an algebraic and from a geometrical point of view.
A metabelian group G acting as automorphism group on a compact Riemann surface of genus g ≥ 2 has order less than or equal to 16(g-1). We calculate for which values of g this bound is achieved and on these cases we calculate a presentation of the group G.