Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups

TL;DR

The paper proves that, given a genus-2 quasi-Fuchsian group $\Gamma$ and a cocompact Kleinian group $G$, one can realize a surface subgroup $H<G$ that is $M$-quasiconformally conjugate to a finite-index subgroup of $\Gamma$ for any $M>1$. It develops a comprehensive framework based on pants decompositions, ideal triangulations, and an inefficiency theory for framed segments to construct good assemblies whose geometry is tightly controlled. Key contributions include (i) construction of a non-separating $(R,m)$-good pants decomposition with large cuff lengths and positive real parts of shears, (ii) counting and Hall-type matching of good pants to form coherent assemblies, and (iii) a distortion-bounded gluing procedure that yields a perfect model close to the assembled surface and a quasiconformal conjugacy to a finite-index subgroup of $\Gamma$. The results extend prior work on surface subgroups (Kahn–Markovic, KW21) to the genus-2 quasi-Fuchsian setting and offer techniques potentially extensible to higher-genus and broader Lie-group contexts, linking geometric control with limit-set dimensions and universal ubiquity phenomena.

Abstract

In this paper, we want to control the geometry of some surface subgroups of a cocompact Kleinian group. More precisely, provided any genus-2 quasi-Fuchsian group $Γ$ and cocompact Kleinian group $G$, then for any $K>1$, we will find a surface subgroup $H$ of $G$ that is $K$-quasiconformally conjugate to a finite index subgroup $F<Γ$.

Figures (6)

• Figure 1: A neighborhood of $C_3$.
• Figure 2: An ideal triangulation of $P_1$ and its corresponding picture in the universal cover $\mathbb H^3$, where $\tilde{C_i}$ is a lift of $C_i$ for $i=1,2,3$ and all these lifts may not be on the same hyperbolic plane.
• Figure 3: A picture for Lemma \ref{['lem5.8']}
• Figure 4: $\mathcal{P}^{(\infty)}$ in Case I.
• Figure 5: $\mathcal{P}^{(\infty)}$(the left one) and $\mathcal{P'}^{(\infty)}$ in Case II.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Definition 4.1
• Definition 4.2
• Definition 4.3
• Lemma 4.4: Lemma 4.10 in LM15
• Lemma 4.5
• proof
• Lemma 4.6: Lemma 4.8 in LM15
• Lemma 4.7: Sum of Inefficiencies Lemma