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We compute the spectral dimension, the dimension of a symmetric random walk,and the Gelfand-Kirillov dimension for compact Vilenkin groups. As a result, weshow 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 specificexamples: the group of $p$-adic integers and the $p$-adic Heisenberg group.Finally, we prove the nonexistence of a natural class of spectral triples onthe group of $p$-adic integers.

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