  This paper shows that every Gromov hyperbolic group can be described by a
finite subdivision rule acting on the 3-sphere. This gives a boundary-like
sequence of increasingly refined finite cell complexes which carry all
quasi-isometry information about the group. This extends a result from Cannon
and Swenson in 1998 that hyperbolic groups can be described by a recursive
sequence of overlapping coverings by possibly wild sets, and demonstrates the
existence of non-cubulated groups that can be represented by subdivision rules.
