We shall present some new results
around another well known conjecture of Zimmer, with Margulis, on commensurated subgroups of arithmetic groups, in which the original strategy involving property (T) and amenability is pursuit. In the talk we shall also try to place it in the considerably more general framework put forward jointly with George Willis a few years ago. Time permitting, we hope to say something about its connection with the most fundamental problem in the theory of harmonic functions on groups -- the dependency of the Liouville property on the (finite, symmetric) measure chosen on the group.
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