Strong automorphism realisation conjecture for hyperbolic triangle groups

Let $(p,q,r)$ be a hyperbolic triple, with $p,q,r\in\mathbb{N}\cup\{\infty\}$. Write $\Delta(p,q,r)$ and $\Delta[p,q,r]$ for the corresponding triangle groups, and let the associated categories be the categories of oriented hypermaps and of all hypermaps of type $(p,q,r)$. A category has the strong automorphism realisation property if every countable group is realised as the automorphism group of a connected object in the category, with the relevant strong realisation multiplicity.

Strong automorphism realisation conjecture. The condition that at least one of $p$, $q$ and $r$ is $\infty$ can be omitted from the countable and strong automorphism realisation assertion: the groups $\Delta(p,q,r)$ and $\Delta[p,q,r]$ of any hyperbolic type, together with their associated categories, have the strong automorphism realisation property.

The preceding theorem establishes the assertion when at least one of $p$, $q$ and $r$ is $\infty$. The conjecture asks for the remaining hyperbolic types and is motivated by results on cocompact triangle groups and their finite-index subgroups.