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Steep uncountable groups

Samuel M. Corson, Alexander Olshanskii, Olga Varghese

Abstract

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural number $n_Y$ for which every element of $G$ may be expressed as a product of length at most $n_Y$ of elements in $Y^{\pm 1}$. In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group $G$ can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in $G$.

Steep uncountable groups

Abstract

We produce a simple group of cardinality which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset there exists a natural number for which every element of may be expressed as a product of length at most of elements in . In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in .
Paper Structure (2 sections, 6 theorems, 6 equations)

This paper contains 2 sections, 6 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem

Key Result

Theorem 1

There exists a simple group $G$ such that

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • Remark 6
  • Lemma 7
  • proof : Proof of Theorem \ref{['maintheorem']}
  • Remark 8
  • ...and 2 more