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Changing topological type of compression bodies through cone manifolds

Alex Elzenaar

TL;DR

This work demonstrates that one can explicitly realize smooth, controlled deformations in the SL$(2,\mathbb{C})$ character variety that pass between discrete deformation spaces via cone-manifold structures. Through explicit parameter spaces for genus two surface groups and related compression-body groups, it constructs a concrete path that begins with a maximally cusped end and ends at a $(1;2)$-compression body, with the deformation carried by cone angles along a singular locus. The central mechanism combines Koebe constructions, explicit matrix parameterizations, and cone-manifold theory to show that topological changes in hyperbolic 3-manifolds can be realized by local, angle-controlled deformations. The approach yields a general principle: for suitable pairs of compression bodies with the same compression end, there exists a cone-deformation path connecting their hyperbolic structures, enriching our understanding of how topology and geometry interrelate in hyperbolic 3-manifolds.

Abstract

Generically, small deformations of cone manifold holonomy groups have wildly uncontrolled global geometry. We give a short concrete example showing that it is possible to deform complete hyperbolic metrics on a thickened genus $2$ surface to complete hyperbolic metrics on the genus two handlebody with a single unknotted cusp drilled out via cone manifolds of prescribed singular structure. In other words, there exists a method to construct smooth curves in the character variety of $ π_1(S_{2,0}) $ which join open sets parameterising discrete groups (quasi-conformal deformation spaces) through indiscrete groups where the indiscreteness arises in a very controlled, local, way: a cone angle increasing along a fixed singular locus.

Changing topological type of compression bodies through cone manifolds

TL;DR

This work demonstrates that one can explicitly realize smooth, controlled deformations in the SL character variety that pass between discrete deformation spaces via cone-manifold structures. Through explicit parameter spaces for genus two surface groups and related compression-body groups, it constructs a concrete path that begins with a maximally cusped end and ends at a -compression body, with the deformation carried by cone angles along a singular locus. The central mechanism combines Koebe constructions, explicit matrix parameterizations, and cone-manifold theory to show that topological changes in hyperbolic 3-manifolds can be realized by local, angle-controlled deformations. The approach yields a general principle: for suitable pairs of compression bodies with the same compression end, there exists a cone-deformation path connecting their hyperbolic structures, enriching our understanding of how topology and geometry interrelate in hyperbolic 3-manifolds.

Abstract

Generically, small deformations of cone manifold holonomy groups have wildly uncontrolled global geometry. We give a short concrete example showing that it is possible to deform complete hyperbolic metrics on a thickened genus surface to complete hyperbolic metrics on the genus two handlebody with a single unknotted cusp drilled out via cone manifolds of prescribed singular structure. In other words, there exists a method to construct smooth curves in the character variety of which join open sets parameterising discrete groups (quasi-conformal deformation spaces) through indiscrete groups where the indiscreteness arises in a very controlled, local, way: a cone angle increasing along a fixed singular locus.

Paper Structure

This paper contains 9 sections, 4 theorems, 17 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Explicit parameter spaces
  3. Genus two surface groups
  4. $(1;2)$-compression body groups
  5. Groups with a maximally cusped end
  6. Deformations through cone manifolds
  7. Conic triangle groups
  8. Example of a deformation
  9. Explicit parameters for $\mathcal{G}$

Key Result

Lemma 1

The group is a geometrically finite function group such that $M = \mathbb{H}^3/B$ is homeomorphic to $S_2 \times (-1,1)$, where one end of $M$ is a compact surface of genus $2$ and where the other end of $M$ is a union of two thrice-punctured spheres.

Figures (7)

  • Figure 1: Standard marking on the genus $2$ surface.
  • Figure 2: The limit set (in black) of the $(1;2)$-compression body group with parameters $\alpha = 2.5+0.2i$, $\beta = 0.5 + 4.2i$, and $\lambda = 0.09 + 0.85i$. We also show (in red) the boundary on the Riemann sphere of a fundamental domain (Ford domain) for the group from which the hyperbolic quotient can be determined by an application of the Poincaré polyhedron theorem maskit.
  • Figure 3: Geometric objects associated to the Koebe group $B$ constructed in \ref{['sec:gpmax']}. The curves labelled in (B) are the loops represented by the corresponding elements, not the projection of the paired sides of the fundamental polygon.
  • Figure 4: The quadrilaterals with three angles $0$ and fourth angle $2\zeta$ for various values of $\zeta$.
  • Figure 5: Our deformation is parameterised by $\theta$, which controls the isometric circles of $Y_1$ as shown.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Remark
  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof