What are some applications of these specific pure math areas? I'm a pure math PhD specialized in hyperbolic topology, doing a research postdoc right now which ends in August of 2018. So I'll be starting to apply for my next job in the next couple of months. I've been thinking about looking into applied math jobs in addition to the usual math research and teaching jobs.
I realize that there are jobs one could get in industry just for holding a PhD in math, but I am interested in jobs where my particular areas of knowledge/interest have potential for important scientific applications. Barring that, I would most likely accept the best teaching or research position offer I got (work which I also enjoy).
Below are some areas of math that I have studied. I've started with some specific topics I've studied closely, then put some more general areas that you could say I work in. My question is: what are (if any) some known applications of the following disciplines from pure mathematics, to the sciences?


*

*Hyperbolic 3-manifolds

*Hilbert-Blumenthal surfaces or Hilbert modular varieties

*Dynamical systems using Hamilton's quaternions

*Quaternion algebras (not just Hamilton's quaternions, but generalized quaternion algebras as defined by Dickson).

*Low-dimensional topology

*Arithmetic manifolds

*Algebraic number theory, especially quadratic fields, lattices, rings of integers

 A: You're not going to find anyone in industry willing to pay you to work on low-dimensional topology (and I say this as someone whose PhD is also in low-dimensional topology). You can certainly find companies who would be interested in the fact that you have a PhD and thus have demonstrated the ability to conduct research, work independently, handle difficult and abstract concepts, etc. You might even find a few companies that think in the fields you mention are nifty, though they won't have anything you can actually work on in them. Quaternions, for example, are used in computer graphics, but the level of what you describe would be overkill for what a company would be willing to pay you for. Frankly, looking for important scientific applications is not something that industry often does, at least in pure math. 
To answer your specific question: The NSA does some nontrivial work in cryptography, which ties into the algebraic number theory point in your list. (That's a very broad subject, though, and the NSA is not exactly forthcoming about its research.) There are some places in industry that might be able to tie one of the points on your list to machine learning (say, face recognition), which is something they're very interested in funding. More broadly, you might find a government research lab that has  something involving, say, low-dimesional topology, but that's probably one particular project rather than a full career. I've heard that biotech companies are interested in topology for various reasons, but that might be difficult to get into without a background in biology. If you want continue doing actual math and math research, your best bet by far is to continue on the academic research track.
A: In most fields of pure mathematics, there aren't a lot of industry options that rely directly on your expertise or involve continuing with your research program.  Instead there are jobs which use the skills you've learned as a researcher: problem solving, abstraction, ability to learn complicated ideas quickly, etc.
There are occasional exceptions: NSA or other cryptography if you work in certain areas in number theory, quantum computing labs for certain areas of topology and algebra, etc.
A: While this does not address the availability of jobs in a specific industry, cosmic topology is an area of applied math to which many of these pure math topics apply.
