Kaplansky’s conjectures and actions on CAT(-1) Spaces

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University of New Brunswick


We 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.