[math-fun] Eigenvalues and Asimov's group-metric problem

If F is the complex numbers, using dist(Id,M) is some suitable function of the eigenvalues of M, such as dist=SUM |log(eigenvalue)|, perhaps works?

--In order for this distance to define a "metric" we would need the triangle inequality lemma that the product AB of two matrices, has F(AB)=SUM |log(eigenvalues of AB)| <= F(A) + F(B). Is this lemma valid? Apparently we need to use a branch cut for log(z) on the negative z axis. For FINITE matrix groups, the eigenvalues always are roots of unity since every group element has finite order, and then I think the lemma is true because of viewing it as a statement about the angular effects of rotation matrices (?).