The absolute of finitely generated groups: II. The Laplacian and degenerate parts

TL;DR

This paper explores the structure of the absolute boundary of finitely generated groups, focusing on the Laplacian and degenerate parts, and establishes connections with Markov processes and the Laplace operator.

Contribution

It characterizes the Laplacian part of the absolute, shows its invariance under certain group extensions, and reduces computations for nilpotent groups to their abelianizations.

Findings

Laplacian part is preserved under some central extensions

Reduction of Laplacian part computation to abelianization for nilpotent groups

Analysis of fundamental examples like free, commutative, and Heisenberg groups

Abstract

The article continues the 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 the Laplacian part and degenerate part and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under tak- ing some central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelinization; 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.