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On what finitely generated (left-orderable) simple groups can know about their subgroups

Arman Darbinyan, Markus Steenbock

TL;DR

The paper investigates how finitely generated left-orderable simple overgroups can retain information about embedded subgroups, addressing algebraic, geometric, and algorithmic data. It develops Frattini and quasi-isometric embedding techniques, including explicit constructions using unrestricted wreath products, to embed countable families into a 2-generated group while preserving left-orderability and key decision problems. For recursively enumerated families with decidable word problem, the authors show that the overgroup can be chosen to be computably left-orderable with decidable word and membership problems, and that left-orderability can be inherited by the overgroup under suitable computability assumptions. A central contribution is the equivalence between computable left-orders on the embedded subgroup and the existence of a finitely generated computably left-ordered simple group into which the subgroup embeds with decidable membership, along with a quasi-isometric extension of Thompson/Higman-type results to families of groups.

Abstract

In this paper, we survey some of the recent advances on embeddings into finitely generated (left-orderable) simple group such that the overgroup preserves algorithmic, geometric, or algebraic information about the embedded group. We discuss some new consequences and also extend some of those embedding theorems to countable classes of finitely generated groups.

On what finitely generated (left-orderable) simple groups can know about their subgroups

TL;DR

The paper investigates how finitely generated left-orderable simple overgroups can retain information about embedded subgroups, addressing algebraic, geometric, and algorithmic data. It develops Frattini and quasi-isometric embedding techniques, including explicit constructions using unrestricted wreath products, to embed countable families into a 2-generated group while preserving left-orderability and key decision problems. For recursively enumerated families with decidable word problem, the authors show that the overgroup can be chosen to be computably left-orderable with decidable word and membership problems, and that left-orderability can be inherited by the overgroup under suitable computability assumptions. A central contribution is the equivalence between computable left-orders on the embedded subgroup and the existence of a finitely generated computably left-ordered simple group into which the subgroup embeds with decidable membership, along with a quasi-isometric extension of Thompson/Higman-type results to families of groups.

Abstract

In this paper, we survey some of the recent advances on embeddings into finitely generated (left-orderable) simple group such that the overgroup preserves algorithmic, geometric, or algebraic information about the embedded group. We discuss some new consequences and also extend some of those embedding theorems to countable classes of finitely generated groups.

Paper Structure

This paper contains 7 sections, 13 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Embeddings into simple groups
  3. Stronger versions
  4. Geometric embeddings of families of groups
  5. Geometric embeddings of countable families of groups
  6. Computability aspects
  7. Proof of Theorem \ref{['thm-new-1']}

Key Result

Theorem 1.1

If $G$ is a countable left-orderable group, then $G$ has a Frattini embedding into a finitely generated left-orderable simple group $H$. Moreover, if $G$ is finitely generated, then the embedding can be chosen to be isometric.

Theorems & Definitions (26)

  • Theorem 1.1: Darbinyan-Steenbock, ourpaper
  • Proposition 1.2: Darbinyan-Steenbock, ourpaper
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 16 more