  In this paper we answer a question of Kaimanovich by characterizing (jointly)
bi-harmonic functions on countable, discrete groups with respect to a
symmetric, generating measure. We also study the peripheral Poisson boundary of
$L(\G)$ with respect to Markov operators arising from symmetric, generating
probability measures on a countable, discrete group $\G$. We solve a recent
conjecture of Bhat, Talwar and Kar regarding peripheral eigenvalues and their
corresponding eigenvectors for such Markov operators, and provide a complete
description of the peripheral Poisson boundary in the aforementioned scenario.
