Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Abstract

Through the glasses of didactic reduction: We consider a (periodic) tessellation $\Delta$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $\Delta$ we mean the process of cutting $M$ along corank-1 tile-faces into finitely many convex polyhedral pieces, and rearranging the pieces to a new tight covering of the tessellation $\Delta$. Such a rearrangement defines a permutation of the (centers of the) tiles of $\Delta$, and we are interested in the group $PI(\Delta)$ of all piecewise isometric rearrangements of $\Delta$. In this paper we offer: a) An illustration of piecewise isometric rearrangements in the visually attractive hyperbolic plane, b) an explanation how this is related to Richard Thompson's groups, c) a chapter on the structure of the group pei$(\mathbb Z^n)$ of all piecewise Euclidean rearrangements of the standard tessellation of $\mathbb R^n$ by unit-cubes, and d) results on the finiteness properties of some subgroups of pei$(\mathbb Z^n)$.