Finite groups are often exemplified as the symmetries of some object. Specifically, they are represented as rotations, translations, and reflections that permute the points of a shape while preserving an identical spatial arrangement. These examples are usually for finite shapes in Euclidean space, however, they could also be for infinite patterns in Euclidean space. For example, $\mathbb Z^2$ describes the translational symmetry of an infinite 2d grid at integer points. The points are permuted, but the structure remains identical if the points aren't labeled. Obtaining the original labeling is the same as returning to the identity.
In some instances it is clearly impossible for a finitely generated group to be describing the symmetry of an object or pattern in Euclidean space. Consider the free group on two generators. No matter what action of Euclidean space is assigned to these two elements, it will not have a symmetry that respects the group structure. Adding transformations of the space should correspond to adding the generators together such that any series of actions that gets to the identity in the group gets back to the identity in the spatial representation. But with free groups, I can't just pick two translation vectors in Euclidean space to assign to these two generators because their geometric behavior would be abelian, and free groups are non-abelian. Also, non-abelian groups in Euclidean space always seem to include rotational symmetries with finite order, whereas the order of free generators is infinite. So, it seems like the free group can't be imbedded in Euclidean space. It can be imbedded in hyperbolic space, however, since hyperbolic space spreads out at the same rate as the generators of this group do. So, at any point on the Caley graph with each point representing a transformation in hyperbolic space from the origin, the tree can be made to be symmetrical, and this is not the case in Euclidean space where the tree would overlap itself. When translating and rotating from one point and direction to another, the labels for the points change, but the geometric structure relative to that point and direction is identical and has a one-to-one correspondence with the structure that the generated elements would have.
So, with that explanation out of the way, my question is this: can every finitely generated group be described as a subgroup of the symmetries of either Euclidean or Hyperbolic space? I know there are groups with sub-exponential growth. How do these fit in? Any kind of uniform pattern would seem to grow too slowly for hyperbolic space and too fast for Euclidean space. If there are finitely generated groups that cannot be imbedded either in Euclidean or Hyperbolic space, then what kind of geometry can they be imbedded in, if any? Sorry if this question is overly lengthy or trivial, but none of the other questions seemed to be grasping at exactly what I am wondering about since they never mentioned hyperbolic geometry and don't seem to be talking about the same kind of representations I'm thinking of. I'm not very familiar with the group-theoretic terminology for this kind of thing so I thought I had better explain what I have in mind in detail.