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

Surajit Biswas, Bipul Saurabh

TL;DR

The article addresses whether spectral dimension, random-walk dimension, and GK dimension vanish for compact Vilenkin groups and provides explicit computations showing they are all zero. It develops the $K$-theory of $C(G)$ via inductive limits over finite quotients, giving concrete generators for $K_0$ in the cases $\mathbb{Z}_p$ and $\mathbb{H}_d(\mathbb{Z}_p)$. It then proves nonexistence results for a natural class of equivariant even spectral triples on $\mathbb{Z}_p$, highlighting obstructions to Connes' program in the $p$-adic setting. Together, the results constrain noncommutative-geometric approaches to CVGs and illuminate the structure of their operator algebras.

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.

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

TL;DR

The article addresses whether spectral dimension, random-walk dimension, and GK dimension vanish for compact Vilenkin groups and provides explicit computations showing they are all zero. It develops the -theory of via inductive limits over finite quotients, giving concrete generators for in the cases and . It then proves nonexistence results for a natural class of equivariant even spectral triples on , highlighting obstructions to Connes' program in the -adic setting. Together, the results constrain noncommutative-geometric approaches to CVGs and illuminate the structure of their operator algebras.

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.
Paper Structure (7 sections, 24 theorems, 94 equations)

This paper contains 7 sections, 24 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dimensional Invariants of Compact Vilenkin Groups
  4. $K$-groups
  5. Generators of $K_0\left(C\left(\mathbb{Z}_p\right)\right)$ as a span of characters
  6. Generators of $K_0\left(\mathbb{H}_d\left(\mathbb{Z}_p\right)\right)$ in terms of Matrix coefficients
  7. On Spectral triples

Key Result

Proposition 2.3

For a compact topological group $G$, in the ergodic $C^*$-dynamical system $\left(C(G),G,\Delta\right)$, we have $\mathcal{A}=\mathcal{O}(G)$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • ...and 42 more