Arithmetic of linear algebraic groups over certain fields of dimension two

Let k be an algebraically closed field of characteristic zero. Let K be either a function field in two variables over k or the fraction field of a 2-dimensional, excellent, strictly henselian local domain with residue field k. We show that linear algebraic groups over such afield K satisfy most properties familiar in the context of number fields: finiteness of R-equivalence, Hasse principle for complete homogeneous spaces. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.