On deformations of hyperbolic 3-manifolds with geodesic boundary

Let M be a complete finite-volume hyperbolic 3-manifold with compact non-empty geodesic boundary and k toric cusps, and let T be a geometric partially truncated triangulation of M. We show that the variety of solutions of consistency equations for T is a smooth manifold or real dimension 2k near the point representing the unique complete structure on M. As a consequence, the relation between deformations of triangulations and deformations of representations is completely understood, at least in a neighbourhood of the complete structure. This allows us to prove, for example, that small deformations of the complete triangulation affect the compact tetrahedra and the hyperbolic structure on the geodesic boundary only at the second order.Comment: This is the version published by Algebraic & Geometric Topology on 23 March 200

The Effect of Hawaii’s Vast Diversity on Racial and Social Prejudices

Food is the universal language of the world, and Hawaiians speak SPAM. Hawaii is the largest consumer of SPAM in the world, with their own signature recipe, as well as an annual SPAM party which over 20,000 people attend. Hawaiian locals cannot get enough of the stuff, consuming more than 5 million pounds year. SPAM is just one of many beloved foods in Hawaii, all of which are from different cultures. Residents have access to Chinese rice and stir fry, Korean kimchi and marinated meats, Japanese sashimi and bento boxes, Portuguese tomatoes and chili peppers, Puerto Rican casseroles and pasteles, Filipino sweet potatoes and adobo, American macaroni salad and hamburgers, and Hawaiian taro and kalua pig. Food is just one aspect of a very mixed culture that borrows food, music, religion, and customs that are used every day. Diversity is not tolerated, but embraced in Hawaii

(Bounded) continuous cohomology and Gromov proportionality principle

Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We prove that at least for "reasonable" spaces the inclusion of Cc(X) in C(X) induces an isomorphism in cohomology, thus answering a question posed by Mostow. We also prove that such isomorphism is isometric with respect to the L^infty-norm on cohomology defined by Gromov. As an application, we discuss a cohomological proof of Gromov's proportionality principle for the simplicial volume of Riemannian manifolds.Comment: An important improvement with respect to the preceding version: we are now able to show that continuous cohomology is isomorphic to singular cohomology even for (a large class of) spaces with non-contractible universal covering. Therefore, the definition of locally bounded Borelian cohomology is not needed any more

Hyperbolic manifolds with geodesic boundary which are determined by their fundamental group

Let M and N be n-dimensional connected orientable finite-volume hyperbolic manifolds with geodesic boundary, and let f be a given isomorphism between the fundamental groups of M and N. We study the problem whether there exists an isometry between M and N which induces f. We show that this is always the case if the dimension of M and N is at least four, while in the three-dimensional case the existence of an isometry inducing f is proved under some (necessary) additional conditions on f. Such conditions are trivially satisfied if the boundaries of M and N are both compact.Comment: 12 pages, 1 figur

Commensurability of hyperbolic manifolds with geodesic boundary

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are compact. We show that M is commensurable with M'. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as above and G is a finitely generated group which is quasi-isometric to the fundamental group of M, then there exists a hyperbolic manifold with geodesic boundary M'' with the following properties: M'' is commensurable with M, and G is a finite extension of a group which contains the fundamental group of M'' as a finite-index subgroup.Comment: 26 pages, 4 figure