What is the dual of a pre-injective map?

Question

In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of Abelian groups we get the following:

Definition. Given a finite discrete Abelian group $G$, an (infinite) index set $X$ and taking the product $G^X$ endowed with the product topology, we say that a continuous endomorphism $\phi:G^X\to G^X$ is pre-injective if the restriction of $\phi$ to $G^{(X)}$ (the subgroup of elements with finite support) is injective.

The Pontryagin-Van Kampen dual of $\phi$ is just an endomorphism $\widehat\phi$ of the discrete group $G^{(X)}$. Can we say that $\phi$ is pre-injective just looking at $\widehat\phi$? In other words, what is the algebraic property of $\widehat\phi$ which corresponds to the pre-injectivity of $\phi$?

Answer 1

Well, in the case X is also a group, your question somehow relates with cellular automata, i.e your map $\phi :G^X\rightarrow G^X$ is continuous and X-equivariant. When X is amenable,$\phi$ is pre-injective if and only if $\phi$ is surjective. Thus, your dual $\widehat{\phi}$ will be injective in this case. It is not true anymore when $X=F_2$, the free group with 2 generators, see section 5.11 of Ceccherini-Silberstein, Coornaert's book "Cellular Automata and Groups" for an example of pre-injective but not surjective cellular automaton over $F_2$.