  We construct finitely generated groups with strong fixed point properties.
Let $\mathcal{X}_{ac}$ 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 $X\in \mathcal{X}_{ac}$, 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 by diffeomorphisms on any smooth manifold in
$\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic
groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary
hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper
subset of which generates a finite $p$-group.
