The structure of MDC-Schottky extension groups

TL;DR

This paper investigates the automorphism groups of certain hyperbolic 3-manifolds with boundary, establishing bounds on their size and characterizing cases where these bounds are attained, especially for manifolds homeomorphic to connected sums of specific handlebodies.

Contribution

The paper determines explicit bounds for the automorphism groups of specific hyperbolic 3-manifolds and identifies cases where these bounds are sharp, extending previous knowledge on automorphism group sizes.

Findings

For genus 2, the automorphism group size is at most 12, attained by dihedral groups.

For genus g ≥ 3, the automorphism group size is strictly less than 12(g-1).

The upper bound of 12(g-1) is not attained for g ≥ 3.

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.