Growth of balls vs growth of spheres in hyperbolic groups

Question

Let $G$ be a finitely-generated group equipped with a word-metric. Let $B_n$ and $S_n$ be the $n^{\mathrm{th}}$-ball and $n^{\mathrm{th}}$-sphere, respectively, with respect to the given metric. Define the following quantities for $G$. $$b=\limsup_{n\to\infty} \frac{\log\left(\left|B_n\right|\right)}{n} \text{ and } s=\limsup_{n\to\infty} \frac{\log\left(\left|S_n\right|\right)}{n}.$$ Then $s$ and $b$ are always finite and, when $G$ grows exponentially, $b$ is positive.

Assume now that $G$ is a non-elementary hyperbolic group (in the sense of Gromov). In this case by a theorem of Coornaert (Theorem 7.2 in [1]) also $s$ is positive.

When I looked at the basic example of $G=F_r$, the free group of rank $2\leq r<\infty$, an elementary calculation shows that $s=b=\log\left(2r-1\right)$. I then wondered about the equality cases of the inequality $s\leq b$ for hyperbolic groups. More precisely, my questions are:

(1) Is it possible that $s=b$ for a finitely-generated hyperbolic group rather than free groups?

(2) What examples we know for finitely-generated hyperbolic groups with $s<b$?

[1] Coornaert, Michel, Patterson-Sullivan measures on the boundary of a hyperbolic space in the sense of Gromov, Pac. J. Math. 159, No. 2, 241-270 (1993). ZBL0797.20029.

Answer 1

The equality $s=b$ holds for all finitely-generated infinite groups, and this does not relate to hyperbolic groups or even to exponential growth. It is a straighforward consequence of the isoperimetric inequality, by which for every finitely-generated infinite group $G$, for every $n$ one has $$\left|B_{n}\right|\leq\left(n+1\right)\left|S_{n}\right|.$$ In fact, for non-amenable groups and in particular non-elementary hyperbolic groups, one can replace $\left(n+1\right)$ in the above inequality by a positive constant depending only on the group.

For a proof of this see for instance Chapter 12.2 and in particular Theorem 12.14 in [2].

[2] Mann, Avinoam, How groups grow., London Mathematical Society Lecture Note Series 395. Cambridge: Cambridge University Press (ISBN 978-1-107-65750-2/pbk). ix, 199 p. (2012). ZBL1253.20032.