TY  - JOUR
A1  - Bieri, Robert
A1  - Sach, Heike
T1  - Groups of piecewise isometric permutations of lattice points, or Finitary rearrangements of tessellations
T2  - Journal of the London Mathematical Society
N2  - 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 of 𝑃𝐼(Δ) 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(ℤ𝑛) of all piecewise Euclidean rearrangements of the standard cubically tessellated ℝ𝑛, and (d) results on the finiteness properties of some subgroups of pei(ℤ𝑛).
Y1  - 2022
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/75216
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-752166
SN  - 1469-7750
N1  - MSC 2020
20F65 (primary), 20J05, 22E40, 20B07, 52C22 (secondary)
VL  - 106
IS  - 3
SP  - 1663
EP  - 1724
PB  - Wiley
CY  - Oxford
ER  -