  Grotzsch proved that every triangle-free planar graph is 3-colorable.
Thomassen proved that every planar graph of girth at least five is 3-choosable.
As for other surfaces, Thomassen proved that there are only finitely many
4-critical graphs of girth at least five embeddable in any fixed surface. This
implies a linear-time algorithm for deciding 3-colorablity for graphs of girth
at least five on any fixed surface. Dvorak, Kral and Thomas strengthened
Thomassen's result by proving that the number of vertices in a 4-critical graph
of girth at least five is linear in its genus. They used this result to prove
Havel's conjecture that a planar graph whose triangles are pairwise far enough
apart is 3-colorable. As for list-coloring, Dvorak proved that a planar graph
whose cycles of size at most four are pairwise far enough part is 3-choosable.
  In this article, we generalize these results. First we prove a linear
isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new
results then follow from the theory of hyperbolic families of graphs developed
by Postle and Thomas. In particular, it follows that there are only finitely
many 4-list-critical graphs of girth at least five on any fixed surface, and
that in fact the number of vertices of a 4-list-critical graph is linear in its
genus. This provides independent proofs of the above results while generalizing
Dvorak's result to graphs on surfaces that have large edge-width and yields a
similar result showing that a graph of girth at least five with crossings
pairwise far apart is 3-choosable. Finally, we generalize to surfaces
Thomassen's result that every planar graph of girth at least five has
exponentially many distinct 3-list-colorings. Specifically, we show that every
graph of girth at least five that has a 3-list-coloring has
$2^{\Omega(n)-O(g)}$ distinct 3-list-colorings.
