Suppose that G is a group of rational cohomological dimension n and that G is of type FP(n) over the integers. Under these hypotheses we show that there is a bound on the orders of finite subgroups of G. This extends a result of P. H. Kropholler, who obtained the same conclusion for G of finite rational cohomological dimension and of type FP(infinity) over the integers.
For each n, there are groups G of type FP(n-1) over the integers and of rational cohomological dimension n for which there is no bound on the orders of finite subgroups.
Pub. Matematiques 45 (2001) 259-264.