  We prove that for the $d$-regular tessellations of the hyperbolic plane by
$k$-gons, there are exponentially more self-avoiding walks of length $n$ than
there are self-avoiding polygons of length $n$. We then prove that this
property implies that the self-avoiding walk is ballistic, even on an arbitrary
vertex-transitive graph. Moreover, for every fixed $k$, we show that the
connective constant for self-avoiding walks satisfies the asymptotic expansion
$d-1-O(1/d)$ as $d\to \infty$; on the other hand, the connective constant for
self-avoiding polygons remains bounded. Finally, we show for all but two
tessellations that the number of self-avoiding walks of length $n$ is
comparable to the $n$th power of their connective constant. Some of these
results were previously obtained by Madras and Wu \cite{MaWuSAW} for all but
finitely many regular tessellations of the hyperbolic plane.
