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RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2$ and tree products

Sebastian Bischof

Abstract

In this paper we prove that the group $U_+$ of an RGD-system of type $(4, 4, 4)$ over $\mathbb{F}_2$ contains a certain tree product as a subgroup. The proof relies on a careful analysis of the action on the associated twin building. This result is part of a larger project and we will use it as an induction start to construct uncountably many RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2$.

RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2$ and tree products

Abstract

In this paper we prove that the group of an RGD-system of type over contains a certain tree product as a subgroup. The proof relies on a careful analysis of the action on the associated twin building. This result is part of a larger project and we will use it as an induction start to construct uncountably many RGD-systems of type over .

Paper Structure

This paper contains 4 sections, 24 theorems, 28 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2$
  4. An application

Key Result

Theorem 1

Let $(W, S)$ be of type $(4, 4, 4)$ and let $(G, (U_{\alpha})_{\alpha \in \Phi})$ be an RGD-system of type $(W, S)$ over $\mathbb{F}_2$. Then the canonical homomorphism $U_{sr} \star_{U_s} V_{\{s, t\}} \star_{U_t} U_{trt} \to G$ is injective.

Figures (7)

  • Figure 1: Illustration of the group $V_R$
  • Figure 2: Illustration of the group $O_R$
  • Figure 3: Illustration of the group $V_{R, s}$
  • Figure 4: Illustration of the group $O_{R, s}$
  • Figure 5: Illustration of the group $H_R$
  • ...and 2 more figures

Theorems & Definitions (62)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 2.1
  • Lemma 2.4: BiConstruction
  • Lemma 2.5: see BiCoxGrowth and BiConstruction
  • Lemma 2.6: BiConstruction
  • Lemma 2.7
  • proof
  • ...and 52 more