Can rep-tiles be wild?

Tai-Man Tang

School of Mathematics and Computational Sciences, Xiangtan University,
Hunan 411105, People's Republic of China

We consider rep-tiles (replication-tiles) in $R^3$, tiles of $R^3$ composed of miniatures of itself. Examples of reptiles in $R^2$ include the fundamental domains of quadratic CNS. A two-sphere is wildly embedded in $R^3$ if it is not of the same embedding type of the standard sphere. A 3-cell is wildly embedded if its boundary is. While wild tiles has been constructed, it seems to be unknown if wild rep-tiles exist. We show that under a certain assumption, a reptile in $R^3$ cannot be a wild cell.