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

TL;DR

This paper determines the finiteness properties of Chevalley groups over polynomial rings over finite fields, revealing new results about their finiteness length depending on the places included in the ring.

Contribution

It provides explicit calculations of the finiteness length of Chevalley groups over rings of Laurent polynomials over finite fields, including previously unknown cases.

Findings

Finiteness length is n-1 if S contains one place.

Finiteness length is 2n-1 if S contains both places.

SL_3(F_q[t,t^{-1}]) is of type F_3 but not F_4.

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.