Groups and Dynamics

For every combinatorial polyhedron P, there is a canonical right-angled ideal polyhedron in hyperbolic space by a very pretty construction, due in various degrees of generality to Koebe, Andreev and Thurston. By right-angledness, the group generated by reflections across its faces is a discrete subgroup of hyperbolic isometries. So it makes sense to ask: which "input" polyhedra P does one get an arithmetic group? We explain the solution we found in joint work with Devlin, Felikson, Kontorovich and Whitehead. The main ingredient is a lot of hands-on manipulation of hyperbolic polyhdra.