It is worthwhile to mention the Von Neumann conjecture for locally compact groups under *"every object of type X encountered in the mathematical literature, except when specifically attempting to construct counterexamples to this, satisfies a certain property P"*

At around 1930, Von Neumann introduced the definition of amenable groups. It was believed until 1980 that *a group is non-amenable if and only if it contains a subgroup isomorphic to $\mathbb{F}_2$*. In 1950s, M.M. Day attached Von Neumann's name to this famous conjecture. The version of Von Neumann's conjecture for locally compact groups is as follows: a locally compact group is non-amenable if and only if it contains a topological subgroup isomorphic to $\mathbb{F}_2$, the free group on two generators with discrete topology. It was not disproven until 1980, at which year, the *Tarski monster* was shown to be a non-amenable group that does not contain a subgroup isomorphic to $\mathbb{F}_2$.

The conjecture still holds for connected Lie groups and (more generally) almost connected locally compact groups. $G$ is said to be almost connected if the factor group $G/G_e$ is compact, where $G_e$ is the connected component of the identity $e\in G$.