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The structure of MDC-Schottky extension groups

Rubén A. Hidalgo

Abstract

Let $M^{0}$ be a complete hyperbolic $3$-manifold whose conformal boundary is a closed Riemann surface $S$ of genus $g \geq 2$. If $M=M^{0} \cup S$, then let ${\rm Aut}(S;M)$ be the group of conformal automorphisms of $S$ which extend to hyperbolic isometries of $M^{0}$. If the natural homomorphism at fundamental groups, induced by the natural inclusion of $S$ into $M$, is not injective, then it is known that $|{\rm Aut}(S;M)| \leq 12(g-1)$. If $M$ is a handlebody, then it is also known that the upper bound is attained. In this paper, we consider the case when $M$ is homeomorphic to the connected sum of $g \geq 2$ copies of $D^{*} \times S^{1}$, where $D^{*}$ denotes the punctured closed unit disc and $S^{1}$ the unit circle. In this case, we obtain that: (i) if $g=2$, then $|{\rm Aut}(S;M)| \leq 12$ and the equality is attained, this happening for ${\rm Aut}(S;M)$ isomorphic to the dihedral group of order $12$, and (ii) if $g \geq 3$, then $|{\rm Aut}(S;M)|<12(g-1)$, in particular, the above upper bound is not attained.

The structure of MDC-Schottky extension groups

Abstract

Let be a complete hyperbolic -manifold whose conformal boundary is a closed Riemann surface of genus . If , then let be the group of conformal automorphisms of which extend to hyperbolic isometries of . If the natural homomorphism at fundamental groups, induced by the natural inclusion of into , is not injective, then it is known that . If is a handlebody, then it is also known that the upper bound is attained. In this paper, we consider the case when is homeomorphic to the connected sum of copies of , where denotes the punctured closed unit disc and the unit circle. In this case, we obtain that: (i) if , then and the equality is attained, this happening for isomorphic to the dihedral group of order , and (ii) if , then , in particular, the above upper bound is not attained.

Paper Structure

This paper contains 23 sections, 14 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kleinian and function groups
  4. MDC-Schottky groups
  5. MDC-Schottky extension groups
  6. MDC-Schottky extension groups of rank zero
  7. MDC-Schottky extension groups of rank one
  8. MDC-Schottky groups and uniformizations
  9. Lifting automorphisms to MDC-Schottky uniformizations
  10. MCD-Schottky structures
  11. Main Results
  12. Structural description of MDC-Schottky extension groups
  13. An application to automorphisms of MDC-Schottky handlebodies
  14. Upper bounds for MCD-Schottky extension groups
  15. Proof of Theorem \ref{['estructura']}
  16. ...and 8 more sections

Key Result

Theorem 1

Let $M$ be an MCD-handledody, with a complete hyperbolic structure on its interior and whose conformal boundary is a closed Riemann surface $S$ of genus $g \geq 2$.

Figures (3)

  • Figure 1: $K=\langle A,B, E\rangle *_{\langle E \rangle}\langle E,C,D\rangle$, $E^{d}=1$
  • Figure 2: $K=\langle A,B,E\rangle$, $E^{2}=1$
  • Figure 3: $K=\langle A, B,E,U\rangle$, $U^{2}=1=E^{d}$

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Theorem 3: Geometrical description of MDC-Schottky extension groups
  • Theorem 4
  • Corollary 1
  • ...and 12 more