Finiteness Properties of Chevalley Groups over the Ring of (Laurent) Polynomials over a Finite Field

2011/12/13 by Stefan Witzel, Witzel, Stefan
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Finite Group Theory Research #math.GR #math.GT #msc:20E42 #msc:20F65 #msc:51E24 #msc:57M07

paper · pdf · doi:10.48550/arxiv.1112.2961

119 pages, 10 figures, v2 with new introduction

arxiv created 2012/09/18 · arxiv updated 2012/09/19

Abstract

In these notes we determine the finiteness length of the groups G(O_S) where G is an F_q-isotropic, connected, noncommutative, almost simple F_q-group and O_S is one of F_q[t], F_q[t^-1], and F_q[t,t^-1]. That is, k = F_q(t) and S contains one or both of the places s₀ and s_∞ corresponding to the polynomial p(t) = t respectively to the point at infinity. The statement is that the finiteness length of G(O_S) is n-1 if S contains one of the two places and is 2n-1 if it contains both places, where n is the F_q-rank of G. For example, the group SL₃(F_q[t,t^-1]) is of type F₃ but not of type F₄, a fact that was previously unknown.

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