Several old questions in group theory ask whether a specific class of finitely generated groups K can be characterized as exactly the collection of such groups embedding a canonical subgroup G. For example, Von Neumann, and independently Day, asked whether all finitely generated non-amenable groups contain 𝔽₂ as a subgroup. Even earlier, Burnside asked whether every finitely generated infinite group contains an element of infinite order, that is a subgroup isomorphic to ℤ. Both questions turned out to have negative answers.

However, one can pose analogous geometric versions of these problems about the Cayley graphs of such groups. In contrast to their group-theoretic counterparts, the geometric versions of both the Von Neumann-Day and Burnside problems have positive solutions: Whyte proved that every finitely generated non-amenable group has a Cayley graph that can be partitioned into 4-regular trees, and Seward showed that every finitely generated infinite group has a Cayley graph that can be partitioned into ℤ-paths. We will discuss these results and sketch a proof of Whyte's theorem, highlighting the roles played by the isoperimetric characterization of non-amenability and Hall's matching theorem.