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The absolute of finitely generated groups. II: The Laplacian and degenerate parts. (English. Russian original) Zbl 1483.20125

Funct. Anal. Appl. 52, No. 3, 163-177 (2018); translation from Funkts. Anal. Prilozh. 52, No. 3, 3-21 (2018).
Summary: The article continues a series of papers on the absolute of finitely generated groups [Eur. J. Math. 4, No. 4, 1476–1490 (2018; Zbl 1403.28014)]. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into two parts, Laplacian and degenerate, and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under taking certain central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelianization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group).

MSC:

20P05 Probabilistic methods in group theory
20D15 Finite nilpotent groups, \(p\)-groups
60J05 Discrete-time Markov processes on general state spaces

Citations:

Zbl 1403.28014
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References:

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