Embeddings into left-orderable simple groups

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.
• 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.