Automorphism group of a normal tiling of the plane

A normal tiling of the plane is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a connected set or the empty set, and all tiles are uniformly bounded. Normality condition disallows tiles that are pathologically long or thin. The automorphism group of a tiling is a bijection of vertices such that each edge and tile is mapped to an edge and tile, respectively. So, the automorphisms extend to homeomorphisms of the plane. I have the following questions:

1. Is the automorphism group of a normal tiling finitely generated?
2. Is the group acts on the plane properly discontinuously?

Do the answers change if we restrict them to semi-regular tilings? We know that fundamental domain of an infinitely generated automophism group has infinitely many edges. But the domain may be the union of infinitely many tiles. The literatures on tilings concerns themselves usually with the existence of tilings with special properties, but not the automorphism group.

1 Answer

1. Take a surface (without boundary but not compact) of infinite genus, and tesselate it by polygons so that there are no non-trivial combinatorial symmetries of the surface. The universal covering of this surface will be a normal tesselation of the plane whose group of symmetries is not finitely generated.

2. Any symmetry that sends one polygon to itself via the identity map will be the identity on the neighbouring polygons too (since it fixes them, one of their edges and the two ends of that edge), so by induction will be the identity on the whole plane. Thus there will be at most $$2d$$ symmetries that send each $$d$$-gon to itself. Similarly there will be at most 4 symmetries that send each edge to itself and at most $$2v$$ symmetries that send each vertex of valence $$v$$ to itself. This doesn't quite prove the proper discontinuity of the action. If you had the extra condition that the topology is such that a subset of the plane is open iff its intersection with each polygon is open then you could deduce that the action is properly discontinuous. (This condition rules out chains of increasingly small polygons.)