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

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 s_0 and s_\infty 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 SL_3(F_q[t,t^{-1}]) is of type F_3 but not of type F_4, a fact that was previously unknown.