# Gromov's criterion for quasi-isometry of finitely generated groups

In his book "Asymptotic invariants of inifinite groups" (page 4) Gromov gives an equivalence for the quasi-isometry of two finitely generated groups: Two finitely generated groups $G_1$ and $G_2$ are quasi-isometric if and only if there exist proper actions of $G_1$ and $G_2$ on some locally compact topological space $X$ such that i) actions commute ii) both actions are cocompact Given such a topolgical space I can prove that $G_1$ and $G_2$ are quasi-isometric. For the latter implication Gromov gives a short draft:

We consider a space $F$ of all maps from $G_1$ to $G_2$ with the pointwise convergence. Then we take the closure X of the $G_1 \times G_2$ - orbit of our $f \in F$.
Could somebody explain what does it mean that a function $h: G_1 \rightarrow G_2$ admits pointwise convergence and where does that $f$ function come from (maybe it is supposed to be a sum of orbits of all functions from F)? Thank you for all your answers.

It's not saying that any particular function "admits pointwise convergence". Rather, it's saying to consider the set $F$ of all functions $h:G_1\to G_2$, and to put the topology of pointwise convergence on this set. This is just the product topology on the set $G_2^{G_1}$ (where I would assume $G_2$ has the discrete topology).
The function $f$ is presumably a quasi-isometry $f:G_1\to G_2$, since you are proving that if such a quasi-isometry exists then there exist actions on a space $X$ with the given properties.