Groups of dynamical origin

    A graph $\Gamma$ is called repetitive if for all $r > 0$ there exists $d > 0$ such that for all $x,y \in \Gamma$, there exists an isomorphic copy of $B_r(x)$ (the ball of radius $r$ centered at $x$) at distance less or equal to $d$ from $y$. \\

Consider two kind of growths functions for $\Gamma$: pick $x_0 \in \Gamma$ and define $f_{x_0}(n) = \# B_n(x_0)$ and $f(x) = \max \left\{f_x(n): x \in \Gamma\right\}$