[Paper Review] Blowing up and down compacta with geometrically finite convergence actions of a group

Yoshifumi Matsuda, Shin-ichi Oguni+1 more|arXiv (Cornell University)|Jan 30, 2012

Geometric and Algebraic Topology25 references8 citations

TL;DR

This paper establishes a characterization of blow-ups and blow-downs for compacta with geometrically finite convergence actions of a countable group: one compactum is a blow-up of another if and only if every maximal parabolic subgroup for the first is parabolic for the second. The key result shows that such equivariant continuous maps exist precisely when the peripheral structures satisfy this inclusion, enabling the construction of uncountably many non-geometrically finite blow-downs from a single geometrically finite action.

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.

Motivation & Objective

• To characterize when a compactum with a geometrically finite convergence action is a blow-up of another such compactum.
• To investigate the relationship between peripheral structures (maximal parabolic subgroups) and equivariant continuous maps between compacta.
• To construct uncountably many compacta with minimal non-elementary convergence actions that are blow-downs of a given geometrically finite action but are not geometrically finite themselves.

Proposed method

• Uses the notion of peripheral structures (maximal parabolic subgroups) associated with convergence actions on compacta.
• Applies the equivalence between geometrically finite convergence actions and proper relatively hyperbolic structures on the group.
• Employs equivariant continuous maps between compacta to define blow-ups and blow-downs.
• Leverages properties of limit sets of parabolic subgroups under equivariant maps to analyze the structure of preimages of parabolic points.
• Utilizes the fact that equivariant maps are uniquely determined by the peripheral structure inclusion.
• Constructs uncountably many blow-downs via a family of parameterized actions indexed by {0,1}^N, preserving the peripheral structure but altering dynamics.

Experimental results

Research questions

• RQ1 When does a compactum X with a geometrically finite convergence action of a countable group G admit a G-equivariant continuous map to another compactum Y with a geometrically finite convergence action?
• RQ2 What is the precise condition on the peripheral structures of X and Y that ensures the existence of such a G-equivariant map?
• RQ3 Can one construct uncountably many compacta with minimal non-elementary convergence actions that are blow-downs of a given geometrically finite action but are not geometrically finite?
• RQ4 How do the limit sets of parabolic subgroups behave under equivariant maps between such compacta?
• RQ5 What role does the peripheral structure play in classifying blow-up and blow-down relations in the context of convergence actions?

Key findings

• A G-equivariant continuous map π: X → Y exists between compacta with geometrically finite convergence actions if and only if every maximal parabolic subgroup of G with respect to X is also parabolic with respect to Y.
• The map π is surjective and uniquely determined by the peripheral structure inclusion H(X) → H(Y).
• For any bounded parabolic point q ∈ Y with stabilizer H, the preimage π⁻¹(q) equals the limit set Λ(H,X) of H acting on X.
• There exists an uncountable family {X_λ}_{λ∈{0,1}^ℕ} of compact metrizable spaces with minimal non-elementary convergence actions of G such that each X_λ is a blow-down of X and has the same peripheral structure H(X).
• Each X_λ in the family is a blow-down of X but is not geometrically finite, demonstrating that geometric finiteness is not preserved under blow-downs.
• The construction relies on the existence of uncountably many non-isomorphic actions with the same peripheral structure, showing that the peripheral structure does not classify actions up to equivariant homeomorphism when geometric finiteness is dropped.