Algebraic groups over free and hyperbolic groups

2025/06/26 by Vincent Guirardel, Guirardel, Vincent, Chloé Perin +1
Mathematics · #Geometric and Algebraic Topology #Advanced Algebra and Geometry #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.21350

Abstract

We define an algebraic group over a group G to be a variety - that is, a subset of G^d defined by equations over G - endowed with a group law whose coordinates can be expressed as word maps. In the case where G is a torsion-free hyperbolic group and the underlying variety is irreducible we give a complete description of all algebraic groups.

Citations

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Related

• Free and Hyperbolic Groups are not Equational
• Algebraically hyperbolic groups
• Diophantine geometry over groups and the elementary theory of free and hyperbolic groups
• Algebraic subgroups of acylindrically hyperbolic groups
• A Proof of the Barsotti-Chevalley Theorem on Algebraic Groups