The Coxeter simplex with symbol is a compact hyperbolic 4-simplex and the related Coxeter group Γ is a discrete subgroup of Isom(H 4). The Coxeter simplex with symbol is a spherical 3-simplex, and the related Coxeter group G is the group of symmetries of the regular 120-cell. Using the geometry of the regular 120-cell, Davis [3] constructed an epimorphism Γ → G whose kernel K was torsion-free, ...

We study a 2-dimensional manifold that is homogeneous acted on by a 3-dimensional Lie group G, and that has a 2-form invariant under G. We show that such a manifold can be realized as a surface in the affine 3-space and list such realizations. Mathematics Subject Classification(2000). 53C30, 53C42, 53C45

This paper presents some finiteness results for the number of boundary slopes of immersed proper π1-injective surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3manifolds we obtain uniform quadratic bounds in g, independent of the 3-manifold.

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M, g) and the Lq,p-cohomology of that manifold. The Lq,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential forms in L(M) modulo the exact forms which are exterior differentials of forms in L(M).

The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then G must be finite—and thus belongs to the well-known list of finite subgroups of O(4).

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M, g) and the Lq,p-cohomology of that manifold. The Lq,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential forms in L(M) modulo the exact forms which are exterior differentials of forms in L(M).

Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metrics g̃ conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with respect to g̃ is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2.

If g is an integer ≥ 2, and M is a closed simple 3-manifold such that π1(M) has a subgroup isomorphic to a genus-g surface group and dimZ2 H1(M ;Z2) ≥ max(3g−1, 6), we show that M contains a closed, incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold such that VolM ≤ 3.08, then dimZ2 H1(M ;Z2) ≤ 5.

We associate to each finite presentation of a group G a compact CWcomplex that is a 3-manifold in the complement of a point, and whose fundamental group is isomorphic to G. We use this complex to define a notion of genus for G and give examples, and also define a notion of ‘closed group’. A group has genus 0 if and only if it is the fundamental group of a compact orientable 3-manifold.

We compute the hessian iddW of the natural metric form W on the twistor space T(M, g) of a 4-dimensional Riemannian manifold (M, g). We then adapt the computations to the case of the twistor space T(M, g,D) of a hyperkähler manifold (M, g,D = (I, J,K)). We show a strong positivity property of the hessian iddW on the twistor space T(M, g,D) and prove, as an application, a convexity property of t...