Is it decidable whether a finite group presentation is diagrammatically aspherical (that is there is no reduced spherical diagram over this presentation)? Probably - not, but I cannot find a reference. 



Asphericity of 2-complexes
The answer is no.


This follows from a theorem of Collins and Miller, who constructed a recursive sequence of presentations $P\_n$ such that the set of $n$ for which $P\_n$ presents the trivial group is recursively enumerable but not recursive, and $P\_n$ is aspherical if and only if it presents a non-trivial group. I'll add a precise citation later today.


