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

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

2012-01-30

arxiv.org/abs/1201.6104v1

Mathematics

Group Theory

19 pages

Scientific paper

We consider two compacta with minimal non-elementary convergence actions of a countable group. When there exists an equivariant continuous map from one to the other, we call the first a blow-up of the second and the second a blow-down of the first. The following are shown: When both actions are geometrically finite, one is a blow-up of the other if and only if each maximal parabolic subgroup with respect to the first is a parabolic subgroup with respect to the second; A compactum with a geometrically finite convergence action of a finitely generated group has countably infinitely many blow-downs with the same set of maximal parabolic subgroups; Any family of compacta with geometrically finite convergence actions of a finitely generated group has the smallest common blow-up, on which the action is geometrically finite if the family is finite.

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