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arXiv · 2505.01439

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

Surajit Biswas·Bipul Saurabh
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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.

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BibTeXRIS

Surajit Biswas, Bipul Saurabh. 2025-04-26. Vanishing of dimensions and nonexistence of spectral triples on compact Vilenkin groups. https://arxiv.org/abs/2505.01439

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