Tameness phrase

In mathematics , the conjecture of tameness is a conjecture going back to Albert Marden from the theory of Klein's groups in 3-dimensional topology, which was proven in 2004 by Ian Agol , Danny Calegari and David Gabai .

statement

Every complete , 3-dimensional hyperbolic manifold with a finitely generated fundamental group is topologically tame , that is, it is homeomorphic to the interior of a compact manifold.

Ends of hyperbolic 3-manifolds

From the topological tameness it follows immediately that every orientable, complete 3-dimensional hyperbolic manifold with a finitely generated fundamental group can be broken down into a compact kernel (which is to be homeomorphic ) and finitely many connected "ends" which are of the form . The surfaces are homeomorphic to the connected components of . ${\ displaystyle M}$ ${\ displaystyle K}$ ${\ displaystyle M}$ ${\ displaystyle S_ {i} \ times (0, \ infty)}$ ${\ displaystyle S_ {1}, \ ldots, S_ {k}}$ ${\ displaystyle \ partial K}$

Role of hyperbolicity

The assumption that it is hyperbolic plays an essential role in proving the conjecture of tameness. There are counterexamples of (non-hyperbolic) 3-manifolds with finitely generated fundamental groups whose ends are not tame. ${\ displaystyle M}$

literature

Ian Agol : Tameness of hyperbolic 3-manifolds. 2004, arxiv : math.GT/0405568 .
Danny Calegari , David Gabai : Shrinkwrapping and the taming of hyperbolic 3-manifolds. In: Journal of the American Mathematical Society . Vol. 19, No. 2, 2006, pp. 385-446, JSTOR 20161283 .
Richard D. Canary : Marden's tameness conjecture: history and applications In: Lizhen Ji, Kefeng Liu , Lo Yang, Shing-Tung Yau (eds.): Geometry, Analysis and Topology of Discrete groups (= Advanced Lectures in Mathematics. 6). International Press et al., Somerville MA et al. 2008, ISBN 978-1-57146-126-1 , pp. 137-162, ( online (PDF; 246 KB) ).
Dana Mackenzie: Taming the hyperbolic jungle by pruning its unruly edges. In: Science . Vol. 306, No. 5705, 2004, pp. 2182-2183, doi : 10.1126 / science.306.5705.2182 .
Albert Marden : The geometry of finitely generated kleinian groups. In: Annals of Mathematics . Series 2, Vol. 99, No. 3, 1974, pp. 383-462, doi : 10.2307 / 1971059 .