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

Abstract

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. When both actions are geometrically finite, it is shown that one is a blow-up of the other if and only if each parabolic subgroup with respect to the first is parabolic with respect to the second. As an application, for each compactum with a geometrically finite convergence action, we construct its blow-downs with convergence actions which are not geometrically finite.

26 pages, v2: Theorem 1.3 improved (see Theorems 1.4 and 1.5), Appendix B added, reference added, v3: organization of the paper modified, Remarks 1.3, 2.4 and 5.9 added, reference added

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Cited by in corpus (7)

• Relations between various boundaries of relatively hyperbolic groups
• On Cannon-Thurston maps for relatively hyperbolic groups
• The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups
• Hyperbolically embedded virtually free subgroups of relatively hyperbolic groups
• The Bowditch boundary of $(G,\mathcal{H})$ when $G$ is hyperbolic
• Similar relatively hyperbolic actions of a group
• On the quasiconvexity of the fixed subgroup of endomorphisms of relatively hyperbolic groups