  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.
