TITLE: Asphericity of 2-complexes
QUESTION [6 upvotes]: 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.

REPLY [9 votes]: 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.