Let $G$ be a finitely generated group and $H\subseteq G$ a finitely generated subgroup. For every finite set of a generators $F$ for $H$, we can extend to a finite generating set $\hat{F}$ for $G$. Denoting the word metrics on $H$ and $G$ with respect to these generating sets by $d_{1}(x,y)$ and $d_{2}(x,y)$, what relationships exist between $d_{1}$ and $d_{2}$. Presumably for all $x,y\in H$ we have that $d_{1}(x,y)\geq d_{2}(x,y)$, but can we say something more specific?
Edit: The word metric on a group $G$ with respect to a finite generating set $F=\{a_{1},\ldots,a_{n}\}$ is defined in the following way. For an element $x\in G$ the natural number $|x|$ is the length of the shortest word in $F$ and inverses of elements thereof that is equal to $x$. The word metric on $G$ with respect to $F$ is defined by setting $d(x,y):=|xy^{-1}|$.
Best Answer
Yes you have $d_1(x,y) \geq \ d_2(x,y)$ in the above set up but in general there is not a ton to say. As mentioned in the comments this is this idea of distortion which measures certain aspects of how the subgroup $H$ fits inside the group $G$. If you would like you can look at this blog post which discusses some of the ideas and defines this distortion functions, which intuitively compares the intrinsic geometry of the subgroup(its own word metric) and how that subgroup fits inside the full group.
A simple example comes from a Baumslag-Solitar group $G=BS(1,2)= \langle a,t \mid tat^{-1} = a^2 \rangle$(discussed in the above blog post, which you should look at -- it has pictures). Consider $H=\langle a \rangle$ in $G$. Well
$$t^nat^{-n}=t^{n-1} t a t^{-1}t^{-n+1}=t^{n-1} a^2 t^{-n+1}= (t^{n-1} a t^{-n+1}) (t^{n-1}a t^{-n+1})= \dots =a^{2^n} $$
which gives that $d_1(1,a^{2^n})=2^n \geq 2n+1 \geq d_2(1,a^{2^n})$ which is a pretty big difference in the geometry.
Now sometimes you can say more, although normally "coarse-ify" things up to some sort of equivalence so that the choice of generating sets does not change the answer. For example in hyperbolic groups and CAT(0) groups it is known that abelian subgroups are undistorted/quasi-isometrically embedded.