Volume estimates for hyperbolic Coxeter polyhedra

This thesis gives a method to estimate the volume of a hyperbolic Coxeter polyhedron in terms of the combinatorics of its 1-skeleton. Andreev's theorem and Steinitz's theorem together give necessary and sufficient conditions for an abstract graph with edges labeled by non-obtuse dihedral angles to be realizable as a finite volume hyperbolic polyhedron, unique up to isometry. This implies that the volume of a non-obtuse hyperbolic polyhedron is determined by the combinatorics of its labeled 1-skeleton.;The main result of this thesis is a lower bound on the volume of any hyperbolic Coxeter polyhedron in terms of its labeled 1-skeleton. The lower bound follows from two-sided combinatorial volume estimates for equiangular Coxeter polyhedra, linear in the number of vertices, and a characterization of the smallest volume Coxeter polyhedron among families of polyhedra which correspond to graph orbifolds when all dihedral angles are replaced by right angles. The ingredients used to deduce the general lower volume bound include techniques which were used to prove Thurston's Orbifold theorem along with Schlafli's formula which describes the variation of the volume of a smooth family of hyperbolic polyhedra.