Let G be a Chevalley group (scheme) defined over ℤ, simple and simply-connected, and A = k[t] the ring of polynomials over a field k. We shall describe an action of the group Γ = G(k[t]) on an appropriate contractible space, and deduce from that information about the presentations and the homology of the group Γ.

REDUCTION THEORY ON BUILDINGS

Let G and A be as above, and call

K = k(t) the fraction field of A, G the group G(K),

ω the valuation defined on K by ω(u/v) = deg v - deg u, 0 the ring of integers for this valuation (0 ≠ A),

T a maximal torus in G, ϕ the set of roots of G with respect to T, and S ⊂ ϕ a set of simple roots,

T the (affine) Bruhat-Tits building associated to G and ω [1],

the standard apartment associated to T, ϕ the vertex fixed by G(0), 2 the ‘quartier’ with vertex ϕ associated to S, e the fundamental chamber containing ϕ,

G ⊂ SLn an imbedding of G in a special linear group such that T is diagonal and r = SLn(A) ∩ G,

j:T → T' an injection of T into the building T' of SLn(K), compatible with the preceding imbedding, mapping into the standard apartment of T' and multiplying the distances by a fixed constant (cf. [1], 9-1-19, c)).