Diffuse actions on trees
by Tom Norton
Abstract: In the 1940s, Kaplansky conjectured that for a torsion free group G and a field K, the group ring KG contains no zero divisors. We will explore the diffuse property for (torsion-free) groups. Informally, a group G is diffuse if every finite subset A of G with |A| > 1 contains (at least) two extreme elements; such elements prohibit KG from containing zero divisors. In particular, we will explore diffuseness in the context of groups acting on trees, and conclude by describing my research to date in generalising this tree argument into the world of groups acting on CAT(0) cube complexes.
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