 
Summary: HYPERBOLIC SURFACES WITH PRESCRIBED
INFINITE SYMMETRY GROUPS
DANIEL ALLCOCK
Abstract. For any countable group G whatsoever, there is a
complete hyperbolic surface whose isometry group is G. The ar
gument is elementary.
Greenberg proved in 1972 [2] that for any finite group G there is a
closed hyperbolic 2manifold having isometry group isomorphic to G.
Kojima proved the corresponding result for hyperbolic 3manifolds [3].
Following work of Long and Reid [4], Belolipetsky and Lubotzky ex
tended this to hyperbolic manifolds of arbitrary dimension [1]. Our
purpose here is to give a generalization in a di#erent direction, which
also provides an elementary proof of Greenberg's original theorem.
Theorem. For any countable group G, there is a complete hyperbolic
surface X having isometry group isomorphic to G. When G is finite,
X may be taken to be closed.
We build X by gluing together pairs of pants. A pair of pants is a
surface homeomorphic to the 2sphere minus three open disks whose
closures are disjoint, equipped with a hyperbolic metric under which the
boundary curves are geodesic. (All boundary curves in this note are as
