The Structure and Interpretation of Cosmology: Part I - General Relativistic Cosmology

The purpose of this work is to review, clarify, and critically analyse modern mathematical cosmology. The emphasis is upon mathematical objects and structures, rather than numerical computations. This paper concentrates on general relativistic cosmology. The opening section reviews and clarifies the Friedmann-Robertson-Walker models of general relativistic cosmology, while Section 2 deals with the spatially homogeneous models. Particular attention is paid in these opening sections to the topological and geometrical aspects of cosmological models. Section 3 explains how the mathematical formalism can be linked with astronomical observation. In particular, the informal, observational notion of the celestial sphere is given a rigorous mathematical implementation. Part II of this work will concentrate on inflationary cosmology and quantum cosmology

Envelopes of certain solvable groups

A discrete subgroup $\Gamma$ of a locally compact group $H$ is called a uniform lattice if the quotient $H/\Gamma$ is compact. Such an $H$ is called an envelope of $\Gamma$. In this paper we study the problem of classifying envelopes of various solvable groups including the solvable Baumslag-Solitar groups, lamplighter groups and certain abelian-by-cyclic groups. Our techniques are geometric and quasi-isometric in nature. In particular we show that for every $\Gamma$ we consider there is a finite family of preferred model spaces $X$ such that, up to compact groups, $H$ is a cocompact subgroup of $Isom(X)$.Comment: To appear in Commentarii Mathematici Helvetici, 30 page

Contracting automorphisms and L^p-cohomology in degree one

We characterize those Lie groups, and algebraic groups over a local field of characteristic zero, whose first reduced L^p-cohomology is zero for all p>1, extending a result of Pansu. As an application, we obtain a description of Gromov-hyperbolic groups among those groups. In particular we prove that any non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local field of zero characteristic is quasi-isometric to a 3-regular tree. We also extend the study to semidirect products of a general locally compact group by a cyclic group acting by contracting automorphisms.Comment: 27 pages, no figur