The Absolute of finitely generated groups: I. Commutative groups

TL;DR

This paper characterizes the absolute of commutative finitely generated groups and semigroups, showing it consists of measures from Markov chains with i.i.d. increments, generalizing de Finetti's theorem.

Contribution

It provides a complete description of the absolute for commutative groups, extending classical results and linking it to measures of maximal entropy and Markov chains.

Findings

Absolute of commutative semigroups equals measures from i.i.d. Markov chains.

Topologically, the absolute forms a finite-dimensional closed disk.

Generalizes de Finetti's theorem to a broader class of groups and semigroups.

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 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.