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

TL;DR

This paper explores groups of piecewise isometric permutations of lattice tessellations in Euclidean and hyperbolic spaces, illustrating their structure, connections to Richard Thompson's groups, and finiteness properties of certain subgroups.

Contribution

It introduces the group of all piecewise isometric rearrangements of tessellations, relates it to Thompson's groups, and analyzes the structure and finiteness properties of subgroups.

Findings

Visualization of rearrangements in hyperbolic plane

Connection to Richard Thompson's groups

Finiteness properties of subgroups of pei(Z^n)

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$(\mathbb Z^n)$.