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Embeddings into left-orderable simple groups.

Arman Darbinyan1, Markus Steenbock2

  • 1Department of Mathematics Texas A&M College Station Texas USA.

Journal of the London Mathematical Society
|August 1, 2022
PubMed
Summary

Every countable left-ordered group can be embedded into a finitely generated simple group, preserving computability. This research extends group theory with new embedding theorems and computable structures.

Area of Science:

  • Group Theory
  • Computability Theory
  • Order Theory

Background:

  • Left-ordered groups are fundamental in abstract algebra.
  • Embeddings preserve algebraic structure and properties.
  • Computable structures are crucial for theoretical computer science and logic.

Purpose of the Study:

  • To demonstrate that every countable left-ordered group embeds into a finitely generated simple group.
  • To investigate the computability of the left-order in the resulting simple group.
  • To establish a Boone-Higman-Thompson type theorem for left-orderable groups.

Main Methods:

  • Utilizing techniques for constructing embeddings into simple groups.
  • Leveraging computability theory to analyze the properties of the embedded groups.

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  • Applying methods from geometric group theory and universal algebra.

    • Main Results:

    • Proved that every countable left-ordered group embeds into a finitely generated left-ordered simple group.
    • Showed that if the initial group has a computable left-order, the simple group also has a computable left-order.
    • Obtained a Boone-Higman-Thompson type theorem for left-orderable groups with recursively enumerable positive cones.
    • Demonstrated that embeddings are Frattini and isometric for finitely generated initial groups.
    • Reproved Thompson's theorem on word-problem-preserving embeddings into finitely generated simple groups.

    Conclusions:

    • The study establishes significant embedding theorems in the theory of left-ordered groups.
    • The results have implications for understanding the relationship between orderability and computability in groups.
    • The work contributes to the broader study of simple groups and their properties.