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We consider two compacta with minimal non-elementary convergence actions of acountable group. When there exists an equivariant continuous map from one tothe other, we call the first a blow-up of the second and the second a blow-downof the first. When both actions are geometrically finite, it is shown that oneis a blow-up of the other if and only if each parabolic subgroup with respectto the first is parabolic with respect to the second. As an application, foreach compactum with a geometrically finite convergence action, we construct itsblow-downs with convergence actions which are not geometrically finite.

Blowing up and down compacta with geometrically finite convergence actions of a group