Geodesic
The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 3-21

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The article continues a series of papers on the absolute of finitely generated groups. 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).
Keywords: absolute, Laplace operator, dynamic Cayley graph, nilpotent groups, Laplacian part of absolute. 
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     author = {A. M. Vershik and A. V. Malyutin},
     title = {The {Absolute} of {Finitely} {Generated} {Groups:} {II.} {The} {Laplacian} and {Degenerate} {Parts}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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A. M. Vershik; A. V. Malyutin. The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 3-21. http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a0/