Infinitely generated groups and Quasi-isometry.

Question

Let $G$ be an infinitely generated group with Cayley graphs $X$. Define the edges of $X$ to be of length $1$. This makes $X$ a metric space.

Is it possible for $G$ to be quasi-isometric to a finitely generated (presented) group?

Any reference, suggestion or comments will be extremely helpful. Thanks in advance.

Answer 1

Certainly. For instance, if you take the generating set to be all of $G$, then $G$ is bounded, and so is quasi-isometric to any finite group. You can get examples where $G$ is quasi-isometric to an arbitrary finitely generated group $H$ by then taking a product with $H$.

(Note that the quasi-isometry class of $G$ depends very much on the set you choose to use as generators. This is in contrast with the finitely generated case, where all finite sets of generators give the same Cayley graph up to quasi-isometry. As far as I know, when $G$ is infinitely generated, there is typically no "canonical" class of generating sets which give quasi-isometric Cayley graphs analogous to the class of finite generating sets in the finitely generated case.)

Answer 2

There is a different definition of "quasi-isometry" for countably generated groups, originally due to Yehuda Shalom (I think originally from this paper), which doesn't reference cayley graphs at all.

We say a function $f\colon G\to H$ is a quasi-isometric embedding (or uniform embedding) if:

1. For every finite $F\subseteq G$, there is a finite $K\subseteq H$ such that for every $x,y\in G$, $xy^{-1}\in F\Rightarrow f(x)f(y)^{-1}\in K$
2. For every finite $K\subseteq H$, there is a finite $F\subseteq G$ such that for every $x,y\in G$, $f(x)f(y)^{-1}\in K\Rightarrow xy^{-1}\in F$
3. There is a finite $C\subseteq H$ such that $f(G)C=H$

If $G$ and $H$ are finitely generated, groups, this definition coincides precisely with the usual notion of quasi-isometry. Some authors prefer this definition even for finitely generated groups, because it makes the independence of generating set obvious, rather than a theorem.

Under this definition, it is impossible for a non-finitely generated group to be quasi-isometric to a finitely generated one.