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On subgroups of Brin-Thompson groups $nV$

Sadayoshi Kojima, Xiaobing Sheng

Abstract

We prove that the Brin-Thompson group $nV$ is torsion locally finite for $ n \geq 1$ which is known only when $n = 1$, and $nV$ contains elements of infinite order admitting roots with arbitrary large order for $n \geq 2$ which is known to not be true for the $n = 1$ case.

On subgroups of Brin-Thompson groups $nV$

Abstract

We prove that the Brin-Thompson group is torsion locally finite for which is known only when , and contains elements of infinite order admitting roots with arbitrary large order for which is known to not be true for the case.
Paper Structure (9 sections, 22 theorems, 66 equations, 7 figures)

This paper contains 9 sections, 22 theorems, 66 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dyadic block
  4. Brin-Thompson group $nV$
  5. Torsion
  6. Power of elements
  7. Elements of finite order
  8. Roots
  9. Future Topics

Key Result

Theorem 1.1

$nV$ is torsion locally finite for $n \geq 1$.

Figures (7)

  • Figure 1: Rescaling map $\varphi_*$
  • Figure 2: $X, Y \; and \; X \wedge Y$
  • Figure 3: The map $\bar{\pi}$
  • Figure 4: $(X, X, id) \; \; and \; \; (X', X', id)$
  • Figure 5: An example of $(X, X, \sigma)$
  • ...and 2 more figures

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Dyadic block
  • Definition 2.6: Refinement
  • Lemma 2.1
  • proof
  • ...and 58 more