Through the glasses of didactic reduction: We consider a (periodic) tessellation ∆ of either Euclidean or hyperbolic n-space M . By a piecewise isometric rearrangement of ∆ 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 ∆. Such a rearrangement defines a permutation of the (centers of the) tiles of ∆, and we are interested in the group P I(∆) of all piecewise isometric rearrangements of ∆.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(Z n ) of all piecewise Euclidean rearrangements of the standard tessellation of R n by unitcubes, and d) results on the finiteness properties of some subgroups of pei(Z n ). ContentsChapter 1. Introduction 3 1. Generalities and main result 3 Chapter 2. On the hyperbolic case 11 2. Planar hyperbolic examples 11 Chapter 3. The Euclidean case I: The structure of pei(S) 20 3. Orthohedral sets 20 4. Permutation groups supported on orthohedral sets 24

Through the glasses of didactic reduction, we consider a (periodic) tessellation Δ of either Euclidean or hyperbolic 𝑛-space 𝑀. By a piecewise isometric rearrangement of Δ 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 Δ. Such a rearrangement defines a permutation of the (centers of the) tiles of Δ, and we are interested in the group 𝑃𝐼(Δ) of all piecewise isometric rearrangements of Δ. In this paper, we offer (a) an illustration of piecewise isometric rearrangements in the visually attractive hyperbolic plane, (b) an explanation on how this is related to Richard Thompson's groups, (c) a section on the structure of the group pei(ℤ 𝑛 ) of all piecewise Euclidean rearrangements of the standard cubically tessellated ℝ 𝑛 , and (d) results on the finiteness properties of some subgroups of pei(ℤ 𝑛 ).