isometric embeddings of Cayley graphs in "nice" spaces This is from a physicist I know and as may be expected, I am threading my way between poorly defined and poorly translated.
What groups have Cayley graphs (w.r.t. a fixed finite generating set, and metrized so that each edge has length 1) that can be isometrically embedded in some reasonably nice space?  Let us start with either f.d. Euclidean space, or f.d. hyperbolic space as the list of reasonably nice spaces.
This fails to mention whether the embedding should be a quasi-isometry, but let us bring in that question only if it seems to be looming in importance.
Yes, Cartesian products of finitely many copies of $\mathbf Z$ did occur to me, as well as (in dimension 2) triangle groups, but the length restrictions will severely restrict which ones work.
Helpful contributions will be either a few examples, or a few families, or pointers to what references might be relevant and anything else that might be relevant.
 A: You can get a general answer avoiding the discussion on the choice of the generating set or of the metric on the larger space: which finitely generated (f.g.) groups have a bilipschitz embedding into a Euclidean space?
The answer is: virtually abelian f.g. groups (i.e. having a f.g. abelian group of finite index).
Indeed such groups admit proper cocompact actions on $\mathbf{R}^n$ for some $n$, giving rise to a quasi-isometric embedding which can easily be deformed to a bilipschitz embedding (alternatively, if you like to stay with an action, you can add some extra dimensions to make the action [whose kernel is finite] faithful but you lose cocompactness).
Conversely, if a group $\Gamma$ admits a bilipschitz embedding (coarse embedding --aka uniform embedding-- would be enough) into a Euclidean space, by a growth argument, the group has polynomial growth. So it is virtually nilpotent, by Gromov's polynomial growth Theorem. Then a result of general result of Pauls (2001) shows that a virtually nilpotent f.g. group admits a bilipschitz embedding into a CAT(0) space (or a uniformly convex Banach space). However, I guess that this follows, in this special case (Euclidean target) from Pansu's 1989 Annals paper about metric differentiability. The argument consists of observing that a bilipschitz map $\Gamma\to\mathbf{R}^n$ induces a bilipschitz embedding from the asymptotic cone (which is a simply connected nilpotent group $G$ with a Carnot-Caratheodory metric, by an older result of Pansu) into $\mathbf{R}^n$, to show that generically it is metrically differentiable and the differential is generically an injective homomorphism; in particular $G$ is abelian, and this holds iff the original discrete group is virtually abelian.
To go back to the original question: if you stick to isometric embeddings, what you get as a corollary is that if $\Gamma$ is a group with a finite generating set $S$, and $E$ is a finite-dimensional Banach space, then if the Cayley graph of $\Gamma$ w.r.t. $S$ embeds isometrically into $E$ then $\Gamma$ is virtually abelian (which is an extremely restrictive condition). The converse, however, depends on the specific choice of $S$ and the norm on $E$, as pointed out in other answers.
A: I'd like to suggest to study the embeddings of Cayley graphs into $\mathbf Z^n$ with the max metrics $d$:
$$d(x \ y) := \max_{k=1...n}\ |x_k-y_k|$$
My short note, on (the degree of) the universality of the $\mathbf R^n$ space with the max distance function (metrics), published in AMS Notices (in the late 1970s I'd think), may motivate the above approach. Let me stress that a finite metric space with integer distances is isometrically embeddable in $\mathbf R^n\quad\Leftrightarrow\quad$ it is embeddable into $\mathbf Z^n$, both spaces considered with the max metrics.
The max metrics makes embeddings easy.
A: In my first answer above I considered the embeddings of the group only (according to the Cayley graph). The requirement to embed isometrically the whole graph (an infinite metric space) in a sense does not add any complication at all when you embed the whole graph into $\mathbf R^n$ with the max metrics, while you may insist on embedding the vertices into $\mathbf Z^n\subseteq\mathbf R^n$  (if you want to).  Indeed, once vertices are embedded, the embedding can be extended to an isometric embedding of the whole graph (injectivity); furthermore, there is exactly one canonical (the simplest) isometric extension.
While considering other distance functions in the Euclidean spaces can be of some interest, it's clear to me that max is special, is "the right one"--"the best". This is also what the theory of categories strongly suggests.
A: The Cayley graph of arbitrary cyclic group, for any $1$-element generating set, can be isometrically embedded in   $R^{\lceil\frac n2\rceil}$   with the distance function given by maximum (while the group itself gets embedded in   $Z^3$).
On the other hand it is not difficult to show that the cyclic group $Z/5$, with a 1-generator presentation, cannot be isometrically embedded in $R^3$ with the max distance.
