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

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