We consider tilings of (subsets of) Euclidean or hyperbolic n-space S. A permutation of the tiles of S is piecewise isometric (pi) if it is given by isometries on the junks of a finite decomposition of S, and we are interested in the group of all pi-permutations pi(S) of S. If S is the Euclidean strip [0,n]x[0,inf) then pi(S) is Chris Hougton's group H(n) on n rays. Based on Ken Brown's comutation of the finiteness length fl(H(n)) we have results on fl(pei(S)) for Euclidean sqare tilings in all dimensions, and we observe that if S is the hyperbolic plane with the standard SL(2,Z)-tiling then phi(S) is closely related to Thompson's group V.