Discontinuous group actions in hyperbolic geometry with application to units in group rings (FWOTM750)

The main idea is to find new methods to describe the group of units in an integral group ring. For a finite group G={g1,g2,g3,...,gn}, the integral group ring is defined as all the linear combinations of the from z1g1+z2g2+...+zngn where the coefficients z1,z2,...,zn can be any integers. The group of units is the group of all invertible elements among those elements. There are methods to establish such invertible elements, but there are some exceptional cases on which those methods do not apply. So the idea of the project is to find new methods which can handle those exceptional cases. One idea is to translate the problem into a problem of geometric nature. Indeed, some of the groups we are interested in, execute transformations on the hyperbolic space of dimension 2, 3, 4 and 5. By studying geometric properties of these transformations, one is able to get a description of the group itself. The main objective of the project is to extend those methods to actions on more complicated spaces, which are constructed by taking products of several hyperbolic spaces. In a first instance, one hopes to be able to handle groups of units, where a description was completely unknown until now. But also, by doing these new examples, there is hope to get new information on units and maybe find new structures which could lead to generic constructions of units.

Keywords:AATO, WISK, DWIS, CAMP, TWIS, ALG, Mathematics

Disciplines:Geometry