John G. Ratcliffe

The size of a hyperbolic Coxeter simplex

We determine the covolumes of all hyperbolic Coxeter simplex reflection groups. These groups exist up to dimension 9. the volume computations involve several different methods according to the parity of dimension, subgroup relations and arithmeticity properties.

The volume spectrum of hyperbolic 4-manifolds

We construct complete, open, hyperbolic 4-manifolds of smallest volume by gluing together the sides of a regular ideal 24-cell in hyperbolic 4-space. We also show that the volume spectrum of hyperbolic 4-manifolds is the set of all positive integral multiples of 47π2/3.

Commensurability classes of hyperbolic Coxeter groups

In this paper, we classify all the hyperbolic Coxeter n-simplex reflection groups up to widecommensurability for all n 3. We also determine all the subgroup relationships among the groups.

Gravitational instantons of constant curvature

In this paper, we classify all closed flat 4-manifolds that have a reflective symmetry along a separating totally geodesic hypersurface. We also give examples of small-volume hyperbolic 4-manifolds that have a reflective symmetry along a separating totally geodesic hypersurface. Our examples are constructed by gluing together polytopes in hyperbolic 4-space.

Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, C)

In this paper, it is shown that every finitely generated 3-manifold fundamental group G has a rational Euler characteristic χ(G). Lower and upper bounds for χ(G) are given in terms of the rank and deficiency of G. It is shown that every finitely generated 3-manifold group G, with χ(G)<0, is SQ-universal, that is, every countable group can be embedded as a subgroup of a quotient of G. It is also shown that every finitely generated discrete subgroup Γ of SL(2, C) has an Euler characteristics, and χ(Γ)⪰0 if and only if either SL(2, C)/Γ has finite invariant volume or Γ is abelian by finite.

On the Davis hyperbolic 4-manifold

Abstract We algebraically characterize the Davis hyperbolic 4-manifold as the orbit space of the unique torsion-free normal subgroup of index 14,400 of the (5,3,3,5) Coxeter simplex reflection group acting on hyperbolic 4-space. We determine the homology, injectivity radius, and the group of isometries of the Davis manifold. We show that the Davis manifold is a spin manifold.

Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure

Many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. Generalizing to dimension 4, we con- struct a dozen examples of noncompact hyperbolic 4-manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of some of those manifolds are then shown to be complements of tori and Klein bottles in other simply-connected closed 4-manifolds. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic 4-manifolds of mini- mal volume. Our examples are finite covers of some of those manifolds. AMS Classification 57M50, 57Q45 Keywords Hyperbolic 4-manifolds, links in the 4-sphere, links in simply- connected closed 4-manifolds

Fibered orbifolds and crystallographic groups

Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure. A geometric fibration of E^n/G corresponds to a space group extension 1 -> N -> G -> G/N -> 1. We give a criterion for the splitting of a space group extension in terms of the structure group action that is strong enough to detect the splitting of all the space group extensions corresponding to the standard Seifert fibrations of a compact, connected, flat 3-orbifold. If G is an arbitrary n-space group, we prove that the group Isom(E^n/G) of isometries of E^n/G is a compact Lie group whose component of the identity is a torus of dimension equal to the first Betti number of G. This implies that Isom(E^n/G) is finite if and only if G/[G,G] is finite. We describe how to classify all the geometric fibrations of compact, connected, flat n-orbifolds, over a 1-orbifold, up to affine equivalence. We apply our classification theory to the scientifically important case n = 3, and classify all the geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.

On the growth of the number of hyperbolic gravitational instantons with respect to volume

In this paper, we show that the number of hyperbolic gravitational instantons grows superexponentially with respect to volume. As an application, we show that the Hartle-Hawking wavefunction for the universe is infinitely peaked at a certain closed hyperbolic 3-manifold.

Matching theorems for systems of a finitely generated Coxeter group

We study the relationship between two sets S and S 0 of Coxeter generators of a finitely generated Coxeter group W by proving a series of theorems that identify common features of S and S 0 . We describe an algorithm for constructing from any set of Coxeter generators S of W a set of Coxeter generators R of maximum rank for W . A subset C of S is called complete if any two elements of C generate a finite group. We prove that if S and S 0 have maximum rank, then there is a bijection between the complete subsets of S and the complete subsets of S 0 so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of .W;S/ and .W;S 0 / have the same multiset of entries.