Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space. Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a cocompact action of $\Gamma$ by isometries on a CAT(-1) space $(X,d')$, $x\in X$ and constants $n,C>0$ such that $$|d(g,h)-nd'(gx,hx)|<C$$ for all $g,h \in \Gamma$?
(I.e., is the word metric $d$ at bounded distance from the distance induced by the orbital embedding $g\mapsto gx_0$ of $\Gamma$ into some CAT(-k) metric space endowed with a geometric $\Gamma$-action?)
Are there any (non-free) examples of $\Gamma$ for which this is always true?
I am particularly interested when $\Gamma$ is the fundamental group of a closed surface.
A motivation for me is to study some fine properties of the Patterson-Sullivan measure for the word metric, whose measure class is not preserved under quasi-isometry but is preserved under the equivalence described in the question.