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We study the potentially undecidable problem of whether a given 2-dimensional CW complex can be embedded into $\mathbb{R}^4$. We provide operations that preserve embeddability, including joining and cloning of 2-cells, as well as $\Delta\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm Y\Delta$-transformations do not preserve embeddability. We use these results to study 4-flat graphs, i.e., graphs that embed in $\mathbb{R}^4$ after attaching any number of 2-cells to their cycles; a graph class that naturally generalizes planarity and linklessness. We verify several conjectures of van der Holst; in particular, we prove that each of the 78 graphs of the Heawood family is an excluded minor for the class of 4-flat graphs.

We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric probability measure $\mu$ on $G$, such that Bernoulli percolation on $G$ with connection probabilities proportional to $\mu$ behaves like a Poisson branching process with parameter 1 in a sense made precise below. We show that $pdim(G)$ has several natural properties: it is monotone decreasing with respect to subgroups and quotients, and coincides with the growth rate exponent for several classes of groups.

Halin's well-known grid theorem states that a graph $G$ with a thick end must contain a subdivision of the hexagonal half-grid. We obtain the following strengthening when $G$ is vertex-transitive and locally finite. Either $G$ is quasi-isometric to a tree (and therefore has no thick end), or it contains a subdivision of the full hexagonal grid., Comment: 12 pages