How does the wider cryptographic community view non-abelian group based cryptography? Is there perhaps some neural expository article on crypto systems based on non-abelian groups?

I've gleaned that Anshel–Anshel–Goldfeld key exchange is the most well-known cryptographic algorithm based upon non-abelian groups.  Do people consider it respectably secure?  Why?  Why not?

Any opinions about the role of combinatorial group theory here? 

I'm curious because, as I understand it, non-abelian finite groups aren't considered particularly useful in cryptography, due to the Coincident Group Orders Theorem that follows from the Classification of the Finite Simple Groups, but obviously infinite non-abelian groups are another matter.
Almost all cryptographic algorithms which use groups actually work in subgroups generated by a conventional element; even if the group as a whole is non-abelian, the subgroup is cyclic, thus abelian. The Anshel-Anshel-Goldfeld protocol tries to use non-commutativity itself, and relies on "how much non-abelian" the group is.

All asymmetric cryptographic algorithms rely at some point on a presumably hard problem. We do not know whether hard problems really exist, neither asymptotically (that's the whole P=NP problem) nor in practice (when given a specific computational budget, e.g. $2^{80}$ elementary operations). Our only tool is accumulation of work: we take many cryptographers and mathematicians, we let them loose on the problem for a few decades, and we see if they find some way to solve the problem.

To be honest, I had never heard of the Anshel-Anshel-Goldfeld protocol before this morning. It is described in an article which was published in Mathematical Research Letters, which is not usually associated with cryptologic research. There is a bit of research on doing cryptography with braid groups (the AAG protocol is just an instance of that); see this page for a lot of relevant pointers. It is still a recent and widely unexplored area, so, regardless of its inherent merits, the AAG protocol cannot be deemed secure yet. Also, it is only a framework, explained generically over an unspecified non-abelian group; it cannot be declared secure in abstracto.



Infinite non-abelian groups, or anything infinite for that matter, have a slight practical issue: they do not fit well in actual computers which have only a finite amount of RAM. Even for infinite sets where each element can be encoded in a finite (but unbounded) amount of RAM, such as plain integers, variable size easily leads to side-channel leakage, and that's a bad thing.