  We give a complete description of the absolute of commutative finitely
generated groups and semigroups. The absolute (previously called the exit
boundary) is a further elaboration of the notion of the boundary of a random
walk on a group (the Poisson--Furstenberg boundary); namely, the absolute of a
(semi)group is the set of ergodic central measures on the compactum of all
infinite trajectories of a simple random walk on the group. Related notions
have been discussed in the probability literature: Martin boundary, entrance
and exit boundaries (Dynkin), central measures on path spaces of graphs
(Vershik--Kerov). A central measure (with respect to a finite system of
generators of a group or semigroup) is a Markov measure on the space of
trajectories whose cotransition distribution at every point is the uniform
distribution on the generators (i.e., a measure of maximal entropy). For a more
general notion of measures with a given cocycle. For the group~$\Bbb Z$, the
problem of describing the absolute is solved exactly by the classical
de~Finetti's theorem. The main result of this paper, which is a far-reaching
generalization of de~Finetti's theorem, is as follows: the absolute of a
commutative semigroup coincides with the set of central measures corresponding
to (nonstationary) Markov chains with independent identically distributed
increments. Topologically, the absolute is (in the main case) a closed disk of
finite dimension.
