Geometric orbifolds

Abstract

An orbifold is a topological space which «locally Iooks like» the orbit space of a properly discontinuous group action on a manifoid. After a brief review of basic concepts, we consider the special case of 3-dimensional orbifolds of the form f'\M, where Al is a simply-connected 3-dimensional homogeneous space correspondiñg to one of Thurston's eight geometries, and where f.c Isom(M) acts properly discontinuously. A general description of these geometric orbifolds is given and the closed oriented geometric 3-orbifolds with S 3 as their underlying topological space are enumerated (except br hyperbolic orbifolds>. U-.1R2/(dihedral group of order 2n) at each comer point labelled «n», and U-*1R2/(reflection) at alí other points on the boundary of the surface (so = {cone points} U OX 2) Any oriented closed 3-orbifold~3 can be specified by giving an oriented 3-manifold (= X 3) containing a trivalent graph (= Sc'), each edge of which is labelled with an integer> 1 such that at each vertex, the labels n 1, ti2, n3 associated to 4w three incident edges satisfy 1/ni + 1¡n2 1980 Mathematics Subject Classification (1985 revision>: 22E40.

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

“…Such anti-de Sitter 3 manifolds had been recently classified by Francois [59]. In spite of the fact that they have received considerable attention in physics literature [3] recently, the complete description of this phase ( which by the way contains all known crystallographic groups and much more [60] ) in physical terms remains challenging research problem. Thus,the results of Refs.…”

On discrete generalised triangle groups

“…We now deal with the four exceptional cases excluded in (i). These groups have been investigated in [15] (see also [3,5]) and the groups F(2,3;2), F(3,3;2) are finite of orders 24, 288 respectively while the groups F(2,3; 3), F(2,4; 2) are infinite and solvable. We see immediately that the second group cannot have a faithful discrete representation in PSL(2,C).…”

On the cyclic coverings of the knot 52

“…As a final step we identify r 3 and r 4 with r 2 and r\ obtaining a threesphere, where the branching set is a knot embedded as in Figure 7.' Hence, M, is homeomorphic to a 3-sphere and the branching set is the two-bridge knot b (7,3) In particular, M n is homeomorphic to the Lins-Mandel space S(n, 7,3, n -1) -S(n, 7,4,1).…”

“…Therefore, H n is a Heegaard diagram of a 3-manifold, which is an n-fold cyclic covering of S 3 with branching set independent of n. A simple test 2 shows that H 2 is a Heegaard diagram of the lens space L (7,3). Since L(7, 3) admits a unique representation as 2-fold branched covering of S 3 (namely, over the two-bridge knot b(7, 3)), we have the following result: …”