Sorry for the dumb hand-wavy question, but I'm looking for a concept but I don't quite know what to look for.
It's sometimes mentioned that a hyperbolic space is in some sense "bigger" than a Euclidean space, but I'm looking for something a bit more solid. In particular, I am looking for a concept (e.g. volume entropy) which shows that, say, a ball $B(r)$ in $H^3$ can "store" more information than a ball in $E^3$.
[edit] Volume Entropy and Information Capacity:
As it's suggested below, we know that volume of a ball $V(B(r))$ in $H^3$ is larger than $V(B(r))$ in $E^3$ for any radius $r$, so I guess we can go ahead and say that the number of states in a given volume is proportional [1] to $V(B(r))$ and its information capacity is equal to its entropy $S = k \ln V$.
If this is true, then the volume entropy $h = \lim_{r \to + \infty} \frac{\ln(V(B(r))}{r}$ is a measure of asymptotic information density of the entire space.
[1] or is it proportional to the surface area? Or does that only apply to black holes? I really don't know anything about information theory or statistical mechanics, so very much appreciate any pointers.
Related Question: Entropy in $S^3$:
(let me know if this should be a new question)
What if we apply the concept of volume entropy to a ball in $S^3$. If $r$ gets too big, then it wraps around again, but we've already covered all the volume in $S^3$ and hence $V(B(r))$ is bounded by the (finite) volume of $S^3$, $V_{max}$. So does this mean we can say, that $h = \lim_{r \to + \infty} \frac{\ln(min[(V(B(r)),V_{max}])}{r} = 0$?
(I'm sure this must be textbook material for cosmology majors who have to think about entropy in AdS, but I have no physics training at all, so appreciate any further advice!)