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

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...

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Bibliographic Details
Published in:European journal of mathematics 2018-12, Vol.4 (4), p.1476-1490
Main Authors: Vershik, Anatoly M., Malyutin, Andrei V.
Format: Article
Language:English
Subjects:
Algebraic Geometry
Entrances
Group theory
Markov chains
Mathematics
Mathematics and Statistics
Random walk
Random walk theory
Research Article
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Summary: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.
ISSN:2199-675X
2199-6768
DOI:10.1007/s40879-018-0263-8