Infinitely many hyperbolic Coxeter groups through dimension 19

Daniel Allcock
University of Texas at Austin and IAS

A Coxeter group means a discrete group generated by reflections, and it has a fundamental domain which is a "Coxeter polytope", meaning that all its angles have the form $\pi/n$. Building the Coxeter group and building the Coxeter polyhedron is essentially the same problem. We are interested in finite-volume Coxeter polytopes in hyperbolic space, and we explain how to build infinitely many examples in every dimension through 19. (No examples at all are known in other dimensions, except for 21.) In most of these dimensions, we even construct exponentially many Coxeter polytopes, meaning that the number of them with volume bounded by $V$ grows at least exponentially with $V$.