  We study self-avoiding walk on graphs whose automorphism group has a
transitive nonunimodular subgroup. We prove that self-avoiding walk is
ballistic, that the bubble diagram converges at criticality, and that the
critical two-point function decays exponentially in the distance from the
origin. This implies that the critical exponent governing the susceptibility
takes its mean-field value, and hence that the number of self-avoiding walks of
length $n$ is comparable to the $n$th power of the connective constant. We also
prove that the same results hold for a large class of repulsive walk models
with a self-intersection based interaction, including the weakly self-avoiding
walk. All these results apply in particular to the product $T_k \times
\mathbb{Z}^d$ of a $k$-regular tree ($k\geq 3$) with $\mathbb{Z}^d$, for which
these results were previously only known for large $k$.
