The odd-sided polygon representation conjecture

Let $G:=\pi_1({\mathcal P})$ be the fundamental group of a $2$-dimensional negatively curved acute polygon ${\mathcal P}$ of finite groups. Theorem asserts that when $n=2k$ is even, $G$ admits a discrete, faithful, convex-cocompact action on a constant-curvature hyperbolic space $\mathbb H^p$, where $p<\infty$ depends on ${\mathcal P}$. Odd-sided polygon representation conjecture. The assertion of Theorem remains valid for all odd $n\ge 5$. The paper explains that the general technique does not work when $n$ is odd, although a discrete, faithful, convex-cocompact representation is constructed for a special class of odd-sided $n$-gons of groups when $n\ge 5$.