Bounded generation of Steinberg groups over Dedekind rings of arithmetic type

2023/07/07 by Kunyavskii, Boris, Lavrenov, Andrei, Plotkin, Eugene +1
#20G07 #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2307.05526

Abstract

The main result of the present paper is bounded elementary generation of the Steinberg groups St(Φ,R) for simply laced root systems Φ of rank ≥ 2 and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded generation of St(Φ,\mathbb F_q[t, t^-1]) for all root systems Φ, and bounded generation of St(Φ,\mathbb F_q[t]) for all root systems Φ≠\mathsf A₁. The proofs are based on a theorem on bounded elementary generation for the corresponding Chevalley groups, where we provide uniform bounds.

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