Computable change in minimum word length of subgroup elements

Question

Let $G$ be an infinite finitely generated group. Fix a finite generating set for $G$.

Define $\mathrm{len}_G:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in the generators and their inverses equal to $g$.

Let $H\subset G$ is an infinite finitely generated subgroup. Fix a finite generating set for $H$.

Question. Under what conditions is there a computable function $m \colon \mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ such that for all $h\in H$ the inequality $$ \mathrm{len}_H(h)\leq m(\mathrm{len}_G(h)) $$ holds?

Answer 1

I'll put my comment here, so that the question has an answer.

The function $\operatorname{len}_H$ is what is called an actual distortion function (for $H$ in $G$) by Margolis, Meakin & Šuniḱ (see [1]). This is a notion that has been studied before in various forms in some papers by e.g. Gromov and Gersten (as @YCor points out). In their paper, MMS prove (Proposition 1.1) the rather straightforward result that as long as the problem of comparing $H$-words with $G$-words is decidable (which is no harder than the word problem for $G$), then a recursive actual distortion function for $H$ in $G$ exists if and only if the membership problem for $H$ in $G$ is decidable. The proof is very short and elementary.

[1] Margolis, Stuart W.; Meakin, John; Šuniḱ, Zoran, Distortion functions and the membership problem for submonoids of groups and monoids., Geometric methods in group theory. Contemporary Mathematics 372, 109-129 (2005). ZBL1103.20049.