On abstract homomorphisms of Chevalley groups over the coordinate rings\n of affine curves

2016/05/10 by Igor A. Rapinchuk, Rapinchuk, Igor A. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1605.03147

openalex publication_date 2016/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to establish a general rigidity statement for\nabstract representations of elementary subgroups of Chevalley groups of rank at\nleast 2 over a class of commutative rings that includes the localizations of\n1-generated rings and the coordinate rings of affine curves. This is achieved\nby developing the approach introduced in our previous work, and in particular\nby verifying condition (Z) over the class of rings at hand. Our main result\nimplies, for example, that any finite-dimensional representation of SL_n(Z[X])\n(for n at least 3) over an algebraically closed field of characteristic 0 has a\nstandard description, yielding thereby the first unconditional rigidity\nstatement for finitely generated linear groups other than arithmetic\ngroups/lattices.\n

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