(Question mildly edited for clarity by request of Matt F.)

If $G$ is a finitely presented group, let $|\cdot|$ denote the word metric with respect to a finite set of generators. Suppose $\nu$ is a finitely supported measure on $G$ with rational weights. A random walk on $G$ is built by at each stage multiplying by an element of $G$, selected independently from $\nu$. The *drift* is defined to be the $\nu^{\mathbb N}$-a.s limit of $|g_1g_2\ldots g_n|/n$.

Does there exist a finitely-presented $G$ and a rational $\nu$ so that $d$ is irrational?