Date of Award

Fall 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Oh, Hee

Abstract

his thesis is a summary of results obtained on papers; the first three are joint with Hee Oh.

Let Γ < PSL2(C) = Isom+(H3 ) be a Zariski dense Kleinian group whose ordinary set Ω = S 2 − Λ has at least two components. Let ρ : Γ → PSL2(C) be a Zariski dense discrete faithful representation with boundary map f : Λ → S 2 on the limit set.

In, Oh and I obtained a new rigidity theorem: if f is conformal on Λ, in the sense that f maps every circular slice of Λ into a circle, then f extends to a Möbius transformation g on S 2 and ρ is the conjugation by g. Moreover, unless ρ is a conjugation, the set Λf of conformal points of f in Λ, defined as the union of every circular slice C ∩ Λ with C a round circle such that f(C ∩ Λ) is contained in a circle, has empty interior in Λ. This can be viewed as a counterpart to Sullivan’s quasiconformal rigidity for finitely generated Kleinian groups with full limit set S 2 , in the complementary setting where Ω is disconnected. We also present its formulation as the rigidity of maps Λ → S 2 sending vertices of every tetrahedron of zero-volume to vertices of a tetrahedron of zero-volume, or preserving real-ness of cross ratios on Λ ⊂ Cb, in relation to Gromov-Thurston’s proof of Mostow rigidity.

The novelty of our proof is a new viewpoint of relating the rigidity of Γ with the dynamics on the higher rank homogeneous space

(id ×ρ)(Γ)\PSL2(C) × PSL2(C)

where we call the higher rank discrete subgroup Γρ := (id ×ρ)(Γ) < PSL2(C) × PSL2(C) the self-joining of Γ via ρ.

Especially when Γ and ρ(Γ) are convex cocompact, Oh and I proved the following dichotomy in:

either Λf = Λ or Hδ (Λf ) = 0,

where Hδ is the δ-dimensional Hausdorff measure with δ := dim Λ. Moreover, in the former case, ρ is a conjugation by a Möbius transformation on S 2 . We also obtain an analogous theorem for geometrically finite subgroups, or more generally for divergence-type Kleinian groups.

Our proof employs higher rank Patterson-Sullivan theory, by introducing the notion of graph-conformal measure on S 2 × S 2 which enables us to connect ergodic theory on the higher rank homogeneous space Γρ\PSL2(C) × PSL2(C) to the rigidity question on ρ : Γ → PSL2(C). As intermediate consequences, we also obtain ergodicity of horospherical foliations, ergodic decompositions of Bowen-Margulis-Sullivan measures and Burger-Roblin measures, and conformal measure rigidity for a certain class of higher rank discrete subgroups. These are results in.

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