bio  website  ilorentz.org/beenakker 

location  Leiden, The Netherlands  
age  
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physicist at Leiden University
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awarded  Enlightened 
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Why is “The Higman Rope Trick” thus named?
well, the "rope trick" refers to a rope which is supported from the wrong direction; the theorem referred to is given at the end of Higman's paper, supporting the whole argument "from the wrong direction" 
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Some calculus in the orthogonal group $O(n)$
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Some calculus in the orthogonal group $O(n)$
deleted "simplistic argument" for the first integral in my answer: the result for $p$ odd is obvious because contributions $\pm g$ to the integral cancel; the result for $p$ even follows without calculation from the fact that only eigenvalues $\pm 1$ of the orthogonal matrix $g$ contribute to the average of the trace (the complex eigenvalues average out to zero); for $n$ odd every $g$ has one such eigenvalue, so the average is one, for $n$ even and ${\rm det}\,g=+1$ there are no such eigenvalues but for ${\rm det}\,g=1$ there are two, so the average is again one. 
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Some calculus in the orthogonal group $O(n)$
@WillSawin  good point, this is not obvious, so I have removed the "simple argument", which was apparently too simple; for the record, it follows after this comment. Thanks for correcting me. 
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awarded  Nice Answer 
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answered  Some calculus in the orthogonal group $O(n)$ 
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Some calculus in the orthogonal group $O(n)$
since the first two integrals are proportional to the unit matrix, you can take the trace to find the coefficient of proportionality and then use theorem 4 from statweb.stanford.edu/~cgates/PERSI/papers/random_matrices.pdf 