  Suppose we are given a homogeneous tree $\mathcal{T}_q$ of degree $q\geq 3$,
where at each vertex sits a lamp, which can be switched on or off. This
structure can be described by the wreath product $(\mathbb{Z}/2)\wr \Gamma$,
where $\Gamma=\ast_{i=1}^q \mathbb{Z}/2$ is the free product group of $q$
factors $\mathbb{Z}/2$. We consider a transient random walk on a Cayley graph
of $(\mathbb{Z}/2)\wr \Gamma$, for which we want to compute lower and upper
bounds for the rate of escape, that is, the speed at which the random walk
flees to infinity.
