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

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 -groups for compact Vilenkin groups. We express the generators of the -groups in terms of the corresponding matrix coefficients for two specific examples: the group of -adic integers and the -adic Heisenberg group. Finally, we prove the nonexistence of a natural class of spectral triples on the group of -adic integers.