Finiteness Properties of Chevalley Groups over a Polynomial Ring over a Finite Field

It is known from work by H.Abels and P.Abramenko that for a classical Fqgroup G of rank n the arithmetic lattice G(Fq[t]) of Fq[t]-points is of type Fn−1 provided that q is large enough. We show that the statement is true without any assumption on q and for any isotropic, absolutely almost simple group G defined over Fq. Let k be a global function field and let G be a connected, noncommutative, absolutely almost simple k-group of positive rank. Let OS be the ring of S-integers in k. For each place p ∈ S, there is an associated euclidean building Xp acted upon by G(kp) ⊇ G(OS). The dimension of the building Xp is the local rank of G at the place p. In [BW07, Theorem 1.2], K.Wortman and the first author have shown that G(OS) is not of type Fd, where d is the sum of local ranks of G at the places in S. This settles the negative part of the following: Rank Conjecture (see [Behr98] or [BW07]). The group G(OS) is of type Fd−1 but not of type Fd. Results in favor of the rank conjecture include [Stuh80] in which U. Stuhler shows that it holds for SL2(OS). This result has been generalized by Wortman and the first author in [BW08] to arbitrary G of global rank one. Concerning higher ranks, H.Abels [Abel91] and P.Abramenko [Abra87] independently proved the rank conjecture for SLn+1(Fq[t]) provided that q is sufficiently large. Abramenko has better bounds, but they still grow exponentially with n. In [Abra96], Abramenko has verified the rank conjecture for G(Fq[t]) for classical groups G again under the hypothesis that q is sufficiently large with a bound depending only on the rank of G. We generalize the last result. Theorem A. Let G be an absolutely almost simple Fq-group of rank n ≥ 1. Then the group G(Fq[t]) is of type Fn−1 but not of type Fn. Most of our argument will be purely geometric, and we shall deduce Theorem A from: