Suppose that we are given some finitely generated group $ G $ and some finite index subgroup of it $ H $. Given a finite generating symmetric generating set $ S \subset G $, we can define the word metric on $ G $:

$$ d_{S}(g,x):= |g^{-1}x|=min(n\in \mathbb{N} : \exists s_{1},...,s_{n}\in S, g^{-1}h=s_{1}\cdot s_{2}\cdot \dots\cdot s_{n}=g^{-1}x) $$ for all $ g,x \in G $

Similarly, take some symmetric generating set $ T \subset H $ and consider the word metric $ d_{T}^{\prime} $ on $H$. It is well known that these are quasi-isometric on H - for some constants $ K>0 , C>0$, we have:

$$ \frac{1}{K} d_S(h,y)-C \le d^{\prime}_{T}(h,y) \le K d_S(h,y)+C \;\;\forall h,y\in H $$

This property is used many times in geometric problems, and it is quite useful.

However, i'm studying the horofunction boundary (especially for nilpotent groups), in which expressions of the form $ d_S(g,x)-d_S(g,x_{0}) $ appear all the time. I want to know how to relate these to expressions of the form $ d_{T}^{\prime}(g,x)-d_{T}^{\prime}(g,x_{0}) $.

This motivates the following questions:

Given a symmetric generating set $ S \subset G $, can one always find a symmetric generating set $T \subset H$ and some $C>0$ such that $$ d_{T}^{\prime}(y,h)-C \le d_{S}(y,h) \le d_{T}^{\prime}(y,h)+C \;\; \forall y,h\in H $$ is satisfied? (Edit: Solved as negative by Michael Stoll - Take $G=\mathbb{Z}$ with $S=\{\pm 1\}$ and $H=2\mathbb{Z}$).

Can one find some symmetric generating sets $ S \subset G $ and $ T \subset H $ and some $C>0$ such that $$ d_{T}^{\prime}(y,h)-C \le d_{S}(y,h) \le d_{T}^{\prime}(y,h)+C \;\; \forall y,h\in H $$ is satisfied?

Does the answer to one of the prior questions change if $H$ is assumed to be nilpotent? (Edit: part 1 is solved by Michael Stoll with the same example above)

Edit: As a response to Michael's example, I would like to raise another question:

- Does the answer to one of the prior questions change if we relax our demand, replacing the wanted inequality by $$ Kd_{T}^{\prime}(y,h)-C \le d_{S}(y,h) \le Kd_{T}^{\prime}(y,h)+C \;\; \forall y,h\in H $$ for some constants $C,K>0$?