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Funktsional. Anal. i Prilozhen., 2018, Volume 52, Issue 3, Pages 3–21 (Mi faa3593)  

This article is cited in 2 scientific papers (total in 2 papers)

The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts

A. M. Vershikabc, A. V. Malyutinab

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Saint Petersburg State University
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: 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.

Funding Agency Grant Number
Russian Science Foundation 17-71-20153


DOI: https://doi.org/10.4213/faa3593

Full text: PDF file (274 kB)
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English version:
Functional Analysis and Its Applications, 2018, 52:3, 163–177

Bibliographic databases:

UDC: 517.9
Received: 21.05.2018

Citation: A. M. Vershik, A. V. Malyutin, “The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts”, Funktsional. Anal. i Prilozhen., 52:3 (2018), 3–21; Funct. Anal. Appl., 52:3 (2018), 163–177

Citation in format AMSBIB
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\paper The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts
\jour Funktsional. Anal. i Prilozhen.
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\issue 3
\pages 3--21
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\yr 2018
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. M. Vershik, A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi)groups”, Eur. J. Math., 4:4 (2018), 1476–1490  crossref  mathscinet  isi  scopus
    2. A. M. Vershik, A. V. Malyutin, “Asymptotic behavior of the number of geodesics in the discrete Heisenberg group”, J. Math. Sci. (N. Y.), 240:5 (2019), 525–534  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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