I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book *Rigid Geometry and Its Applications*. It claims that all quasi $p$-group $G$ could be characterized as certain Galois group of a Galois covering $Y $to $\mathbf{P}\_1$, only ramified at infinity (over algebraic closed field with positive character).
Since many grandmasters have researched this conjecture, what is the importance of Abhyankar's conjecture ? 
Well, of course it relates to the [inverse Galois theory](https://en.wikipedia.org/wiki/Inverse_Galois_problem)...



About Abhyankar's conjecture
For a bit of context, recall that the affine line over an algebraically closed field of characteristic zero is simply connected in the sense that it has no nontrivial étale covers. (You can see this using Riemann-Hurwitz or by reducing to the corresponding topological statement over $\mathbb{C}$.) In characteristic $p>0$ this is false! For example, $\mathbb{A}^1\_k$ has Artin-Schreier covers. This raises the question of what $\pi\_1(\mathbb{A}^1\_k)$ looks like. By construction, this is an inverse limit of finite groups. The weaker question of what these finite groups are is completely answered by the (solution to) Abhyankar's conjecture: these are precisely the quasi-$p$ groups (groups generated by the Sylow $p$-subgroups). I think that this is a pretty remarkable outcome.


