Kaimanovich and Vershik described certain finitely generated groups
of exponential growth such that simple random walk on their Cayley graph
escapes from the identity at a sublinear rate, or equivalently, all bounded
harmonic functions on the Cayley graph are constant. Here we focus on a key
example, called $G_1$ by Kaimanovich and Vershik, and show that
inward-biased random walks on $G_1$ move outward faster than
simple random walk. Indeed, they escape from the identity at a linear rate
provided that the bias parameter is smaller than the growth rate of $G_1$.
These walks can be viewed as random walks interacting with a dynamical
environment on $\mathbb{Z}$. The proof uses potential theory to analyze a
stationary environment as seen from the moving particle.