Non-Abelian Cut Constructions and Hyperk\"ahler Modifications

We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperk"ahler manifolds: one modifies the topology significantly, the other gives metric deformations. We highlight ways in which the geometry of moment maps for non-Abelian hyperk"ahler actions differs from the Abelian case and from the non-Abelian symplectic case.