The absolute of finitely generated groups: I. Commutative (semi)groups

Abstract

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 all ergodic probability measures on the compactum of all infinite trajectories of a simple random walk which has the same so-called cotransition probability as the simple random walk. Related notions have been discussed in the probability literature: Martin boundary, entrance and exit boundaries (Dynkin), central measures on path spaces of graphs [see Vershik (J Math Sci 209(6):860–873, 2015)]. 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 Markov chains with independent identically distributed increments. Topologically, the absolute is (in the main case) a closed disk of finite dimension.