A geometric group action is an isometric group action on a metric space that is proper and cocompact. For
a discrete group 
, properness means that only finitely many 
satisfy 
for any given compact subset 
. Cocompactness means that translates of one compact subset cover the space. For example, 
acts geometrically on 
by integer translations.
A proper action can have a finite kernel, since properness gives finite point stabilizers. For a torsion-free group these stabilizers are trivial. The Cannon conjecture asks for a geometric action on three-dimensional hyperbolic space from a topological condition on the Gromov boundary of a hyperbolic group.
