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Title:
Mini-course. Self-similarity of groups and random walks: Part IV
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Abstract:
Endowing a group with a probability measure is akin to endowing a smooth manifold with a Riemannian metric. The arising Markov chains (random walks on groups) are analogous to the Brownian motion on manifolds. I will discuss the interplay of random walks with self-similar presentations of groups (provided by actions on homogeneous rooted trees with special properties). This relationship turns out to be useful for understanding the Liouville property of groups (absence of non-constant bounded harmonic functions) and amenability (the presence of an invariant mean). No preliminary knowledge of the subject is required.
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