Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / a / Solution

For a locally finite discrete metric space $(X,d)$ with basepoint $x_0$, the growth function of a discrete metric space isIn a Cayley graph, left multiplication by $g{x}_0^{-1}$ is a graph isometry carrying $x_0$ to any other vertex $g$. It therefore gives a bijection $B(x_0,r)\to B(g,r)$ for every $r$. Thus the growth function of a fixed Cayley graph is exactly independent of the basepoint.