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The article continues the series of papers on the absolute of finitelygenerated groups. The absolute of a group with a fixed system of generators isdefined as the set of ergodic Markov measures for which the system ofcotransition probabilities is the same as for the simple (right) random walkgenerated by the uniform distribution on the generators. The absolute is a newboundary of a group, generated by random walks on the group. We divide theabsolute into the Laplacian part and degenerate part and describe theconnection between the absolute, homogeneous Markov processes, and the Laplaceoperator; prove that the Laplacian part is preserved under tak- ing somecentral extensions of groups; reduce the computation of the Laplacian part ofthe absolute of a nilpotent group to that of its abelinization; consider anumber of fundamental examples (free groups, commutative groups, the discreteHeisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph,nilpotent groups, Laplacian part of absolute.

The absolute of finitely generated groups: II. The Laplacian and degenerate parts