TY - JOUR

T1 - The absolute of finitely generated groups: I. Commutative (semi)groups

AU - Vershik, Anatoly M.

AU - Malyutin, Andrei V.

N1 - A.M.Vershik, A.V.Malyutin, The absolute of finitely generated groups: I. Commutative (semi)groups. European Journal of Mathematics, December 2018, Volume 4, Issue 4, pp 1476–1490, https://doi.org/10.1007/s40879-018-0263-8

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Absolute

KW - Central measure

KW - Characters

KW - Topology of absolute

KW - Absolute

KW - Central measure

KW - Characters

KW - Topology of absolute

KW - RANDOM-WALKS

KW - BOUNDARY

UR - http://www.scopus.com/inward/record.url?scp=85055087873&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/absolute-finitely-generated-groups-icommutative-semigroups

U2 - 10.1007/s40879-018-0263-8

DO - 10.1007/s40879-018-0263-8

M3 - Article

VL - 4

SP - 1476

EP - 1490

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 4

ER -