  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.
