Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds

Casella, Alex; Luo, Feng; Tillmann, Stephan

doi:10.1090/proc/13290

Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds
HTML articles powered by AMS MathViewer

by Alex Casella, Feng Luo and Stephan Tillmann PDF

Proc. Amer. Math. Soc. 145 (2017), 3543-3560 Request permission

Abstract:

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.

References

Benjamin A. Burton and Ryan Budney and William Pettersson and others, Regina: Software for 3-manifold topology and normal surface theory, http://regina.sourceforge.net/, 1999–2014.
Young-Eun Choi, Positively oriented ideal triangulations on hyperbolic three-manifolds, Topology 43 (2004), no. 6, 1345–1371. MR 2081429, DOI 10.1016/j.top.2004.02.002
David Futer and François Guéritaud, From angled triangulations to hyperbolic structures, Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math., vol. 541, Amer. Math. Soc., Providence, RI, 2011, pp. 159–182. MR 2796632, DOI 10.1090/conm/541/10683
Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375 (German). MR 141106, DOI 10.1007/BF02559591
Feng Luo, Volume optimization, normal surfaces, and Thurston’s equation on triangulated 3-manifolds, J. Differential Geom. 93 (2013), no. 2, 299–326. MR 3024308
Walter D. Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic $3$-manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 243–271. MR 1184415
Walter D. Neumann and Alan W. Reid, Rigidity of cusps in deformations of hyperbolic $3$-orbifolds, Math. Ann. 295 (1993), no. 2, 223–237. MR 1202390, DOI 10.1007/BF01444885
Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI 10.1016/0040-9383(85)90004-7
Henry Segerman and Stephan Tillmann, Pseudo-developing maps for ideal triangulations I: essential edges and generalised hyperbolic gluing equations, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 85–102. MR 2866925, DOI 10.1090/conm/560/11093
Stephan Tillmann, Normal surfaces in topologically finite 3-manifolds, Enseign. Math. (2) 54 (2008), no. 3-4, 329–380. MR 2478091