Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields

Let G be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by Colliot-Thélène, Gille and Parimala. Basing on their results, we compute the group of classes of R-equivalence G(k)/R, the defect of weak approximation AΣ(G), the first Galois cohomology H1(k,G), and the Tate–Shafarevich kernel ш1(k,G) (for suitable k) in terms of the algebraic fundamental group π1(G). We prove that the groups G(k)/R and AΣ(G) and the set ш1(k,G) are stably k-birational invariants of G.