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Algebraic and topological properties of Riordan groups over finite fields

Gi-Sang Cheon, Nhan-Phu Chung, Minh-Nhat Phung

Abstract

In this paper, we investigate algebraic and topological properties of the Riordan groups over finite fields. These groups provide a new class of topologically finitely generated profinite groups with finite width. We also introduce, characterize index-subgroups of our Riordan groups, and finally we show exactly the range of Hausdorff dimensions of these groups. The latter results are analogous to the work of Barnea and Klopsch for the Nottingham groups.

Algebraic and topological properties of Riordan groups over finite fields

Abstract

In this paper, we investigate algebraic and topological properties of the Riordan groups over finite fields. These groups provide a new class of topologically finitely generated profinite groups with finite width. We also introduce, characterize index-subgroups of our Riordan groups, and finally we show exactly the range of Hausdorff dimensions of these groups. The latter results are analogous to the work of Barnea and Klopsch for the Nottingham groups.

Paper Structure

This paper contains 5 sections, 16 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Riordan groups and their properties
  3. Profiniteness of ${\mathcal{R}}({\mathbb K})$
  4. Lower central series of ${\mathcal{R}}({\mathbb K})$
  5. index-subgroups and Hausdorff dimensions

Key Result

Theorem 1.1

Let ${\mathbb K}$ be a finite commutative unital ring, $p\geq 2$ be a prime number and ${\mathbb F}_q$ be a finite field with characteristic $p$. Then

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • ...and 27 more