Finitely displaced sub-IAPS

Definition
Let $$(G,\Phi)$$ be an IAPS of groups and $$H$$ a sub-IAPS. We say that $$H$$ is $$n$$-displaced for some positive integer $$n$$ if for every positive integer $$m$$, there is an injective homomorphism:

$$G_m \to H_{m+n}$$

such that the restriction to $$H_m$$, viz:

$$H_m \to H_{m+n}$$

is the same as $$g \mapsto \Phi_{m,n}(g,e)$$, viz concatenating with the identity element.

A finitely displaced sub-IAPS is a sub-IAPS that is $$n$$-displaced for some positive integer $$n$$.

Stronger properties

 * Abelian-quotient sub-IAPS