The Structure of the Three-Dimensional Special Linear Group over a Local Field

2018/01/10 by David J. Wright, Wright, David
Mathematics · #05E45 #13H99 #20E06 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1801.03575

openalex publication_date 2018/01/10 · openalex created_date 2018/01/26 · openalex updated_date 2026/07/28

Abstract

For K a local field, it is shown that SL3(K) acts on a simply connected two dimensional simplicial complex in which a single face serves as a fundamental domain. From this it follows that SL3(K) is the generalized amalgamated product of three subgroups. Specifically if K is the field of fractions of the discrete valuation ring O, then SL3(K) is the amalgamation of three subgroups isomorphic to SL3(O) along pairwise intersections. This generalizes a theorem of Ihara, which gives the structure of SL2(K) as the amalgamated product of two groups in analogous fashion.

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