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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.

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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.

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    $\begingroup$ D. Collins, Ch. Miller III. The word problem in groups of cohomological dimension 2. Groups St. Andrews 1997 in Bath, I, 211–218, London Math. Soc. Lecture Note Ser., 260, Cambridge Univ. Press, Cambridge, 1999. MR link: mathscinet.ams.org/mathscinet-getitem?mr=1676618 $\endgroup$
    – YCor
    Commented Dec 3, 2017 at 23:17

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