Jørgensen's algebraic-geometric limits conjecture

Let Γ\Gamma be a finitely generated, torsion-free, non-elementary Kleinian group, and let {ρn}\{\rho_n\} be a sequence in D(Γ)\mathcal D(\Gamma) converging to a representation ρ\rho. Suppose that {ρn(Γ)}\{\rho_n(\Gamma)\} converges geometrically to a Kleinian group Γ^\widehat\Gamma, and that ρ\rho is type-preserving. Jørgensen's conjecture. Then

ρ(Γ)=Γ^.\rho(\Gamma)=\widehat\Gamma.

This conjecture concerns when algebraic and geometric limits agree. The source attributes it to Jørgensen and provides surrounding partial results, but no resolution status for the full statement.

Sources & referencesView supporting material

Primary source

James W. Anderson, “A brief survey of the deformation theory of Kleinian groups”, arXiv:math/9810186 (1998).

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