Vanishing of dimensions and nonexistence of spectral triples on compact Vilenkin groups

TL;DR

This paper calculates various dimensions for compact Vilenkin groups, showing they are zero, and demonstrates the nonexistence of spectral triples on the p-adic integers, providing explicit K-group descriptions.

Contribution

It establishes that spectral, random walk, and Gelfand-Kirillov dimensions vanish for compact Vilenkin groups and proves no natural spectral triples exist on p-adic integers, with explicit K-group descriptions.

Findings

Dimensions are zero for all compact, totally disconnected, metrizable groups.

Explicit K-group generators are described for p-adic integer and p-adic Heisenberg groups.

Spectral triples do not exist on the p-adic integers within the considered class.

Abstract

We compute the spectral dimension, the dimension of a symmetric random walk, and the Gelfand-Kirillov dimension for compact Vilenkin groups. As a result, we show that these dimensions are zero for any compact, totally disconnected, metrizable topological group. We provide an explicit description of the $K$-groups for compact Vilenkin groups. We express the generators of the $K_0$-groups in terms of the corresponding matrix coefficients for two specific examples: the group of $p$-adic integers and the $p$-adic Heisenberg group. Finally, we prove the nonexistence of a natural class of spectral triples on the group of $p$-adic integers.