Infinite groups with fixed point properties

We construct finitely generated groups with strong fixed point properties. Let Xac be the class of Hausdorff spaces of finite covering dimension which are mod-p acyclic for at least one prime p. We produce the first examples of infinite finitely generated groups Q with the property that for any action of Q on any X ∈ Xac, there is a global fixed point. Moreover, Q may be chosen to be simple and to have Kazhdan’s property (T). We construct a finitely presented infinite group P that admits no non-trivial action on any manifold in Xac. In building Q, we exhibit new families of hyperbolic groups: for each n ≥ 1 and each prime p, we construct a non-elementary hyperbolic group Gn,p which has a generating set of size n + 2, any proper subset of which generates a finite p-group.