Group actions on twin buildings

Though parts of the present paper perfectly look like abstract general nonsense, its origin is a rather concrete question. Given a simply connected almost simple Chevalley group G and a field k, what is a fundamental domain (in an appropriate sense) for the action of G(k[t, t−1]) on the product ∆+ × ∆− of the two associated Bruhat–Tits buildings ∆+ = ∆(G(k((t−1)))) and ∆− = ∆(G(k((t)))) ? The solution of this problem is of interest if one wants to determine the finiteness properties of the S–arithmetic group G(IFq[t, t−1]) by similar methods as used in [A1] and [A2]. An answer to the above question is given in Section 3, Proposition 5 below. I first derived this result by applying Theorem 1 of [So] which describes a simplicial fundamental domain (in the strictest sense) for the action of G(k[t]) on ∆+ together with Lemma 4 of Section 3. Then the proof of Lemma 4 also yielded a preliminary version of Lemma 2 involving both affine buildings ∆+ and ∆−. Thus I was led to considering twin buildings and twin BN–pairs which turned out to constitute the most natural framework for the original problem. For example, Soulé’s result can easily be deduced and generalized in this context (see Proposition 6 and Remark 8). As I discovered afterwards, this possibility is already indicated in [T2], §15.5 and §15.7. The action of G(k[t, t−1]) on ∆+ × ∆− provides an example for a group acting “strongly transitively” (cf. Definition 3) on a twin building. In Section 2, we shall study arbitrary actions of this type in an abstract framework. In particular, we shall see that they always admit certain easily describable fundamental domains (cf. Proposition 3 and its corollaries). However, these results can be derived under assumptions on the pair (∆+,∆−) which are weaker than requiring it to be a twin