Infinite groups with fixed point properties

TL;DR

This paper constructs finitely generated groups with universal fixed point properties on certain topological spaces, including simple groups with Kazhdan's property (T), and introduces new hyperbolic groups with specific subgroup structures.

Contribution

It provides the first examples of infinite finitely generated groups with fixed point properties on a broad class of spaces, including simple groups with property (T), and constructs new hyperbolic groups with unique subgroup characteristics.

Findings

Existence of infinite finitely generated groups with fixed point properties on mod-p acyclic spaces.

Construction of simple groups with Kazhdan's property (T) that have fixed point properties.

Development of new hyperbolic groups with generating sets where proper subsets generate finite p-groups.

Abstract

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.