Martin Kneser's work on quadratic forms and algebraic groups.

A general reference is the following:ko,p,k p ,o pG ⊂ GL(V )a totally real algebraic number fieldas beforesimply connected semisimple, almost simple over kWilliam M. Kantor: Finite geometries via algebraic affine buildings,pp. 37-44 in: Finite Geometries, Buildings and Related Topics (Eds.W. M. Kantor et al.), Oxford University Press, Oxford 1990Other contributors: Timmesfeld, Stroth, Ronan, Meixner.We maintain the general notation introduced preciously; in particular,we consider the following:anisotropic at the infinite placesp a fixed finite place of k s.t. rk p G ≥ 2¯k p := o/po the residue field at p∆ := ∆(G(k p )) the Bruhat-Tits building of G(k p ).La lattice in V s.t. o p L =: L p defines a vertex of ∆∆ 0∼ = ∆(G(¯kp )) the residue (star, link) of L in ∆.Γ := G(o[ 1 p ]) a {p}-arithmetic discrete subgroup of G(k p)Γ 0 := G(o)the finite stabilizer of L in G(k).4142The following unpublished result grew out of discussions betweenWilliam M. Kantor, <strong>Martin</strong> Kneser and myself in the early 90s.Proposition (M. Kneser, unpublished) Under the aboveassumptions, the following properties of the lattice L (resp. thearithmetic groups Γ, Γ 0 ) are equivalent:• Γ acts chamber transitively on ∆.• (i) Γ 0 acts chamber transitively on ∆ 0 ,(ii) h G (L) = 1.Proof: “=⇒”: (i) is obvious from the assumption, since thechambers of ∆ 0 are exactly the chambers of ∆ containing the vertexL. For (ii), we have to show thatG(A k ) = G(k) · G(A(∞)). (1)Since G is isotropic at p, we can use strong approximation for the setof places ∞ ∪ {p}:G(A k ) = G(k) · G(A(∞ ∪ {p})). (2)Since Γ acts chamber transitively on ∆ it acts also vertex transitivelyon the vertices of a given type, which for “type L” translates asG(k p ) = Γ · G(o p ) = G(o[ 1 p ]) · G(o p). (3)4344