Suppose $G$ is a finitely generated group, $f: \mathbb{N} \longrightarrow \mathbb{R_+}$ is a function satisfying $1<\frac{f(n)}{f(n+1)}<\lambda$ and $\sum f(n)<\infty$, which we called a floyd scaling function. Then if we consider a finite generating set $S$ for $G$, then we can consider the floyd compactification of $G$, denoted by $\bar{G_f}$. I know that the fact if $G$ is a hyperbolic group and $f(n)=\frac{1}{n^2+1}$, then its floyd boundary is homeomorphic to its Gromov boundary. But I want to prove the fact rigorusly that the floyd boundary of a genus $g\geq 2$ is non-trivial i.e. contains infinitely many points. But don't know how to proceed.
Further it will be helpful if anyone suggest some good literature on floyd compactification. Thanks