This paper is unpublished work of Geoffrey Mess written in 1990, which gives a classification of flat and anti-de Sitter domains of dependence in 2+1 dimensions.
Every finitely generated group r can be endowed with the word metric d (invariant under left translations) with respect to a given finite generating set.Following Gromov [Gr] we say that r is negatively curved provided there exists a (large) number t5 such thatfor all x, y, Z E r. Here (x· y) stands for the "overlap function"Gromov's article [Gr] When r has torsion, we can construct a highly connected polyhedron with free and cocompact r-action. For integers d and m, let P(d, m, f) be the subcomplex of the m-fold join r * ... * r consisting of those simplices whose vertices are within d in r. The natural map P(d, m, f) ---+ Pd(f) has the property that the preimage of each simplex is (m -2)-connected. It follows that for large d, P(d, m, f) is an (m -2)-connected free r-complex with compact quotient. This shows for example that r is a group of type F P 00 • Furthermore, for any ring R (in practice R = Z or Q) the compactly supported cohomology H: (P(d, m, r); R), which is (by definition, see [Br]) isomorphic
Abstract.We give an example of a 4-dimensional Kleinian group which is finitely generated but not finitely presented, and is a subgroup of a cocompact Kleinian group.
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