Teichmüller spaces, polynomial loci, and degeneration in spaces of algebraic correspondences

Abstract

We develop an analog of the notion of a character variety in the context of algebraic correspondences. It turns out that matings of certain Fuchsian groups and polynomials are contained in this ambient character variety. This gives rise to two different analogs of the Bers slice by fixing either the polynomial or the Fuchsian group. The Bers-like slices are homeomorphic copies of Teichmüller spaces or combinatorial copies of polynomial connectedness loci. We show that these slices are bounded in the character variety, thus proving the analog of a theorem of Bers. To produce compactifications of the Bers-like slices, we initiate a study of degeneration of algebraic correspondences on trees of Riemann spheres, revealing a new degeneration phenomenon in conformal dynamics. There is no available analog of Sullivan's 'no invariant line field' theorem in our context. Nevertheless, for the four times punctured sphere, we show that the compactifications of Teichmüller spaces are naturally homeomorphic.

Figures (9)

• Figure 3.1: Pictured is the preferred fundamental domain $\Pi_0$ and the action of the associated Bowen-Series map for a sphere with four punctures.
• Figure 3.2: The dynamical plane of the conformal mating $F$ and the correspondence $\mathfrak{C}$, for some $P\in\mathcal{H}_5$ and $\Gamma\in\mathrm{Teich}(S_{0,4})$, are displayed. The rational map $R$, that mediates between the $\mathfrak{C}$-plane and the $F$-plane, is injective on the interior of the blue curve (marked as $\partial\mathfrak{D}$).
• Figure 4.1: Pictured is the domain and codomain of a degenerate anti-polynomial-like map. Here, $P_1$ (respectively, $P_2$) is a pinched polygon with two pinched points and two corners (respectively, one pinched point and one corner).
• Figure 4.2: Some vertical fibers and horizontal leaves in the total space $\mathfrak{T}$ are depicted. For $P_1,P_2\in\mathcal{H}_{2d-1}$, the corresponding horizontal leaves are precisely the Bers slices $\mathfrak{B}(P_1), \mathfrak{B}(P_2)$. Lack of continuity of the bijective holonomy map $\Psi_\Gamma:\mathscr{B}_{\Gamma_0}\to\mathscr{B}_\Gamma$ represents discontinuity of straightening maps in holomorphic dynamics, and possible failure of continuous boundary extension of the homeomorphism $\chi:\mathfrak{B}(P_1)\to\mathfrak{B}(P_2)$ portrays the Kerckhoff-Thurston discontinuity phenomenon for Kleinian groups.
• Figure 7.1: Illustrated are the two change of coordinates $L_n$ and $M_n$ appearing in Lemma \ref{['lem:tworescalinglimits']}. On the top right figure, the exterior of the blue curve is the domain $\mathfrak{D}_n$.

Theorems & Definitions (103)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Lemma 2.6: Existence of rescaling limits