Groups of piecewise isometric permutations of lattice points

TL;DR

This paper explores groups of piecewise isometric permutations of lattice points in Euclidean and hyperbolic spaces, revealing connections to known groups like Thompson's V and analyzing their finiteness properties.

Contribution

It introduces the concept of piecewise G-isometric permutation groups on lattice orbits, providing basic examples and analyzing their algebraic and finiteness properties.

Findings

Piecewise hyperbolic groups relate to Richard Thompson's group V.

Piecewise Euclidean groups exhibit diverse finiteness properties.

The paper presents foundational examples of these permutation groups.

Abstract

Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in M, we consider the orbit Gp in M and define a notion of "G-polyhedral pieces" S in Gp. The objects of our interest are the groups pi(S) of all piecewise G-isometric permutations on S. In this paper we merely present the two most basic examples, and these play rather different roles: The case when G = PSL(2,Z) acting on the hyperbolic plane reveals that the "piecewise hyperbolic" groups phi(Gp) here have prominent relatives: they are closely related to Richard Thompson's group V. And in the Euclidean case when G = Isom(Z^n) we find that the "piecewise Euclidean" groups pei(S) - as well as the corresponding "piecewise translation" groups pet(S) - have divers but to some extent accessible finiteness properties.