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

A group G is of type F_n if there is a K(G,1) complex that has finite n-skeleton. The property F_1 is equivalent to being finitely generated and the property F_2 is equivalent to being finitely presented. The finiteness length of G is the maximal n for which G is of type F_n if it exists and is infinite otherwise. A rich source of groups with finite finiteness length consists of S-arithmetic groups in positive characteristic, that is, groups of the form H(O_S) where H is an algebraic group defined over a global function field k and O_S is the ring of S-integers for a finite set S of places of k. In this thesis we determine the finiteness length of the groups H(O_S) where H 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_∞ corresponding to the polynomial p(t) = t respectively to the point at infinity. The statement is that the finiteness length of H(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 H. 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.