It's an obvious consequence of the pigeonhole principle that any injective function over finite sets is bijective. But there are some similar results in different areas of mathematics that apply to less-finite settings.


In algebraic geometry, the [Ax–Grothendieck theorem](http://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem) states (if I have it correctly) that any injection from an algebraic variety over an algebraically closed field to itself is bijective; the standard proof involves some sort of local-global principle together with the same fact over finite fields.


In the theory of cellular automata, the [Garden of Eden theorem](http://en.wikipedia.org/wiki/Garden_of_Eden_%28cellular_automaton%29) states that any injective cellular automaton (over an integer grid of some fixed finite dimension, say) is bijective; the standard proof involves again the same fact for finite sets of cells together with a limiting argument that shows that for large enough bounded regions of an unbounded grid, the boundary of the region has negligible effect compared to the interior.


Is there some way of viewing these three injective-bijective statements (or others) as instances of a single more general phenomenon?



Ax–Grothendieck and the Garden of Eden
See the recent paper ["On algebraic cellular automata"](http://www.arxiv.com/abs/1011.4759), for a proof of how to derive the Garden of Eden theorem from the Ax-Grothendieck theorem. This is indeed in the spirit of Gromov's paper that Mohan mentioned in the comments. This paper is were he introduced Sofic groups. The story as I understand it is: The Ax-Grothendieck theorem tells us that every regular algebraic map from a complex algebraic variety to itself can not be a strict embedding. If one takes a power of the variety indexed by a group G, then the analogous result holds for G-equivariant pro-regular maps when the group G satisfies some conditions (can be approximated by "nice" groups). Gromov calls this property "surjunctive". The original GOE theorem is about $G=\mathbb{Z}^n$. It turns out that this works for any amenable group, and more generally for sofic groups, those groups that can be approximated by amenable groups.


