Exotic group actions on homology 3-spheres
Brad Safnuk · cIRcle (University of British Columbia) · 2009
We present two contrasting methods for constructing group actions on homology 3-spheres. These actions are characterized as being exotic - not conjugate to a standard linear action. The first method is to modify a standard linear action on the 3-sphere by performing surgery on an invariant link. A family of exotic actions is produced for a class of metacyclic groups. This method is certainly applicable to any group admitting a linear representation. The second method is a computer aided study using the program Twiggy, software written by the author. Starting from a knot in the 3-sphere, a representation is found from the knot group to a finite group G. The branched cover associated to the representation has a natural G-action. The program looks for representations, determines if the cover is a homology sphere and tests if the action constructed is exotic. The primary test of whether or not an action is exotic is by examining the linking numbers of the fixed point curves.