Uncountably many quasi-isometric torsion-free groups

Key Points

• The aim is to demonstrate the existence of uncountably many finitely generated, pairwise non-isomorphic torsion-free groups within the same quasi-isometry class.
• Utilized Schur covering groups to define group structures.
• Applied principles of group cohomology to analyze properties.
• Incorporated bounded-valued cohomology for geometric insights.
• Constructed uncountably many pairwise non-isomorphic torsion-free groups.
• All constructed groups belong to the same quasi-isometry class.
• Demonstrated new relationships among finitely generated groups through cohomological methods.