TY - JOUR

T1 - The Absolute of Finitely Generated Groups

T2 - II. The Laplacian and Degenerate Parts

AU - Vershik, A. M.

AU - Malyutin, A. V.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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).

AB - 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).

KW - absolute

KW - dynamic Cayley graph

KW - Laplace operator

KW - Laplacian part of absolute

KW - nilpotent groups

UR - http://www.scopus.com/inward/record.url?scp=85055094742&partnerID=8YFLogxK

U2 - 10.1007/s10688-018-0225-4

DO - 10.1007/s10688-018-0225-4

M3 - Article

AN - SCOPUS:85055094742

VL - 52

SP - 163

EP - 177

JO - Functional Analysis and its Applications

JF - Functional Analysis and its Applications

SN - 0016-2663

IS - 3

ER -