The Yagita invariant of general linear groups

We give a definition of the Yagita invariant po(G) of an arbitrary group G, and compute po(GL n (O)) for each prime p, where O is any integrally closed subring of the complex numbers ℂ (i.e., O is integrally closed in its field of fractions F, which is also a subring of ℂ). We also show that po(SL n (O)) is equal to po(GL n (O)) for n ≥ 2 except that possibly po(SL n (O)) is 1/2po (GL n (O)) for some small n and ‘small’ rings O. Our definition of po(G) extends both Yagita’s original definition for G finite [9] and the definition given by one of us for G of finite virtual cohomological dimension (or vcd) [8]. For G of finite vcd such that the Tate-Farrell cohomology Ĥ* (G) of G is p-periodic, po(G) is equal to the p-period. Hence our results may be viewed as a generalisation of those of Biirgisser and Eckmann [1,2], who compute the p-periods of GL n (O) and SL n (O) for various O such that these groups have finite vcd and for all n such that these groups are p-periodic. The methods we use are similar to those used in [1].