  Spin foam models, loop quantum gravity and group field theory are discussed
as quantum gravity candidate theories and usually involve a continuous Lie
group. We advocate here to consider quantum gravity inspired models with finite
groups, firstly as a test bed for the full theory and secondly as a class of
new lattice theories possibly featuring an analogue diffeomorphism symmetry. To
make these notes accessible to readers outside the quantum gravity community we
provide an introduction to some essential concepts in the loop quantum gravity,
spin foam and group field theory approach and point out the many connections to
lattice field theory and condensed matter systems.
