The structure of MDC-Schottky extension groups

Let $M^0$ be a complete hyperbolic $3$-manifold whose conformal boundary is a closed Riemann surface $S$ of genus $g\ge 2$. If $M=M^0\cup S$, then let $\mathrm{A}\mathrm{u}\mathrm{t}(S;M)$ 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 $\mathrm{\mid }\mathrm{A}\mathrm{u}\mathrm{t}(S;M)\mathrm{\mid }\le 12(g-1)$. 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 $g\ge 2$ copies of $D^{\ast }\times S^1$, where $D^{\ast }$ denotes the punctured closed unit disc and $S^1$ the unit circle. In this case, we obtain that: (i) if $g=2$, then $\mathrm{\mid }\mathrm{A}\mathrm{u}\mathrm{t}(S;M)\mathrm{\mid }\le 12$ and the equality is attained, this happening for $\mathrm{A}\mathrm{u}\mathrm{t}(S;M)$ isomorphic to the dihedral group of order $12$, and (ii) if $g\ge 3$, then $\mathrm{\mid }\mathrm{A}\mathrm{u}\mathrm{t}(S;M)\mathrm{\mid }<12(g-1)$, in particular, the above upper bound is not attained.