  The article continues the series of papers on the absolute of finitely
generated groups. The absolute of a group with a fixed system of generators is
defined as the set of ergodic Markov measures for which the system of
cotransition probabilities is the same as for the simple (right) random walk
generated by the uniform distribution on the generators. The absolute is a new
boundary of a group, generated by random walks on the group. We divide the
absolute into the Laplacian part and degenerate part and describe the
connection between the absolute, homogeneous Markov processes, and the Laplace
operator; prove that the Laplacian part is preserved under tak- ing some
central extensions of groups; reduce the computation of the Laplacian part of
the absolute of a nilpotent group to that of its abelinization; consider a
number of fundamental examples (free groups, commutative groups, the discrete
Heisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph,
nilpotent groups, Laplacian part of absolute.
