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We extend several techniques and theorems from geometric group theory so thatthey apply to geometric actions on arbitrary proper metric ARs (absoluteretracts). A second way that we generalize earlier results is by eliminatingfreeness requirements often placed on the group actions. In doing so, we allowfor groups with torsion. The main theorems are new in that they generalizeresults found in the literature, but a significant aim is expository. Towardthat end, brief but reasonably comprehensive introductions to the theories ofANRs (absolute neighborhood retracts) and Z-sets are included, as well as amuch shorter short introduction to shape theory. Here is a sampling of thetheorems proved here.  THEOREM. If quasi-isometric groups G and H act geometrically on proper metricARs X and Y , resp., then X is proper homotopy equivalent to Y.  THEOREM. If quasi-isometric groups G and H act geometrically on proper metricARs X and Y , resp., and Y can be compactified to a Z-structure for H, then thesame boundary can be added to X to obtain a Z-structure for G.  THEOREM. If quasi-isometric groups G and H admit Z-structures (X^, Z_1) and(Y^, Z_2) resp., then Z_1 and Z_2 are shape equivalent.

Proper homotopy types and Z-boundaries of spaces admitting geometric group actions