Timeline for Relative Poincaré 2-complexes
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 29, 2018 at 13:03 | comment | added | HJRW | ... By the Bestvina--Mess theorem (which relates the Cech cohomology of the boundary to the group cohomology), you can deduce the classification of PD_2 groups in the hyperbolic case. The 3-dimensional analogue of the Convergence Group Theorem is the Cannon Conjecture, which is probably the most important component of the question of whether all PD_3 groups are 3-manifold groups. | |
| Mar 29, 2018 at 13:00 | comment | added | HJRW | It's true that the relative case follows easily from the absolute case. (Double to obtain an absolute PD_2 group, which is then a surface, and use the fact that any splitting of a surface over a cyclic subgroup is induced by a simple closed curve.) To my mind, the main geometric thread that follows from this is the work on the Gromov (and Bowditch) boundaries of (relatively) hyperbolic groups. The Convergence Group Theorem of Tukia, Casson--Jungreis and Gabai asserts that a hyperbolic group with circle boundary is commensurable to a surface group. Tukia also did the relative case. (cont'd) | |
| Mar 29, 2018 at 9:35 | comment | added | Ulrik Buchholtz | Thanks @HJRW, for bringing that up. It predates the papers on the absolute case (from 1980 and 1983), but it states (p. 314): “In fact it can be proved that the absolute PD²-conjecture implies the relative one.” This is then sketched for one-relator groups. I think this amplifies my question: was this stuff ever revisited later? (And perhaps from a more geometric/homotopical perspective?) | |
| Mar 29, 2018 at 7:42 | comment | added | HJRW | I believe the reference for the relative statement is Bieri—Eckmann, Relative homology and Poincare duality for group pairs, 1978. | |
| Mar 28, 2018 at 16:06 | history | asked | Ulrik Buchholtz | CC BY-SA 3.0 |