A Gieseking-féle hiperbolikus sokaság "deformációi" ELHALASZTVA!

2020. 03. 17. 10:30
Molnár Emil

In joint work with István Prok and Jenő Szirmai (1999) we determined all Dehn surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold, leading to compact orbifolds S(k), k = 2, 3, … with singularity geodesics of rotation order k. We computed also the volume of S(k), tending to 0 (zero) as k converges to infinity. As the reviewer of Math. Rev., Kevin P. Scannell remarked this is in conflict with the well-known theorem of D. A. Kazhdan and G.A. Margulis [Mat. Sb. (N.S.) 75 (117) (1968), 163-168; MR0223487] and with the work of Thurston, describing the geometric convergence of orbifolds under large Dehn fillings. Now we stand before refreshing our publication, since many authors try to determine this „positive” lower bound that the „theorem” indicates. In dimension 2 there is such an lower bound, this is the area π/42 of hyperbolic triangle of angles π/2, π/3, π/7 (of curvature -1).