Jørgensen's algebraic-geometric limits conjecture

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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.

References

Primary source

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

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