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In fact, there is numerical evidence that the points in this walk 'stay much closer together' than in a random walk. Thus one might be bold to conjecture that even a 3-dimensional version of the prime race is surjective, even though the simple random walk in 3 dimensions is not recurrent. But heuristics are the one thing, rigorous proofs are the other ... .
The examples given so far are of the form $(A \times B^\infty, B^\infty)$, where $A$ both embeds into $B$ and is an epimorphic image of $B$. -- Can anyone give an example where the groups do not admit a decomposition into infinitely many direct factors, or which are at least not of the form $(A \times B^\infty, B^\infty)$?
@Eric: Your analysis seems right to me. -- As the simple random walk on $\mathbb{Z}^m$ is recurrent if and only if $m \leq 2$, this means that $\sum_{n = 1}^\infty F(n)$ diverges if and only if $m \leq 2$ -- correct?
Indeed. -- However which one is 'simpler' depends on your notion of 'simplicity': in order to define a free group, you need no relations, while to define a finite cyclic group, you need 1!
