Combinatorial and Computational aspects of Tilings

One of the classical problems in the theory of crystallographic
structures, like tilings and lattices, is determining the possible
symmetry groups of such structures. In this context, colour
symmetries of crystallographic tilings were studied, for instance by
Belov, Shubnikov, Gr\"unbaum, Senechal... The usual
questions to address are (i) the possible number of colours for a
given tiling, (ii) the number of colour symmetries for a given number
of colours within this tiling (multiplicity), and (iii) the algebraic
structure of the colour group. This has been carried out intensively
for crystallographic tilings in Euclidean space.
In this talk, some answers to all (i), (ii) and (iii) are presented
for hyperbolic regular tilings and hyperbolic Laves tilings. First, the
questions are formulated in the language of finitely presented
groups. This yields in principle a method to compute the number of
colour symmetries of any regular or Laves tiling for a given number of
colours. Moreover, some certain cases are discussed in detail, aided
by a lot of illustrations.