Jointly bi-harmonic functions on groups and Peripheral Poisson boundaries

Assistant Professor Sayan Das; Department of Mathematics, Embry-Riddle Aeronautical University

The study of asymptotic properties of a random walk on a countable, discrete group G (with respect to a symmetric, generating probability measure) relies on understanding a natural boundary of the random walk, called the Poisson boundary. The study of Poisson boundaries is intimately related with the study of bounded harmonic functions on groups. The startling "double ergodicity theorem" of Kaimanovich states that (separately) bi-harmonic functions on groups is constant—a feature that has led to the discovery of many rigidity properties of group representations and group actions. This led Kaimanovich to ask about the characterization of (jointly) bi-harmonic functions. In this talk I will completely characterize bi-harmonic functions on groups, thereby answering Kaimanovich's question, that he posed in 1992.  I will also introduce the notion of "Peripheral Poisson boundaries," which was first considered in a recent paper of Bhat, Talwar and Kar (2022). I will completely characterize the peripheral Poisson boundaries of groups, thereby answering a recent question of Bhat, Talwar and Kar.