Brannock, Matthew2024-03-062024-03-062023-11https://unbscholar.lib.unb.ca/handle/1882/37744We provide specific conditions on a ring R and a group G under which the group ring RG will satisfy the Kaplansky Conjectures on the existence of non-trivial units, zero-divisors and idempotents in the group ring. We give a chain of implications on properties that a group must have to satisfy these conjectures. Specifically, we define a Bowditch action of a group on a type of metric space called a CAT(-1) space and show this action will be spherically diffuse. We then prove that if a group acts on a metric space in a spherically diffusely, then the group itself must be diffuse. Next we prove that if a group is diffuse then it satisfies the Unique Product Property. We then prove that if a group satisfies this property, then the group ring formed by this group and any integral domain will satisfy the Kaplansky Conjectures.vii, 82electronicenhttp://purl.org/coar/access_right/c_abf2master thesisTouikan, NicholasMathematics and Statistics