×

zbMATH — the first resource for mathematics

A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds. (English) Zbl 1151.57014
Authors’ summary: We prove the convex combination theorem for hyperbolic \(n\)-manifolds. Applications are given both in high dimensions and in three dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic \(n\)-space, satisfying a natural condition on their parabolic subgroups and intersection with a separable subgroup, there are finite index subgroups which generate a subgroup that is an amalgamated free product. Constructions of infinite volume hyperbolic \(n\)-manifolds are described by gluing lower \(H\)-dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multiple immersed boundary slope. If the fundamental group of a hyperbolic 3-manifold contains a maximal surface group not carried by an embedded surface, then it contains a freely indecomposable group with second Betti number at least 2.

MSC:
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Agol, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. 153 ((2)) pp 599– (2001) · Zbl 1067.20067
[2] Baker, On boundary slopes of immersed incompressible surfaces, Ann. Inst. Fourier (Grenoble) 46 pp 1443– (1996) · Zbl 0864.57015
[3] Baker, Immersed, virtually-embedded, boundary slopes, Topology Appl. 102 pp 239– (2000) · Zbl 1002.57043
[4] Bestvina, A combination theorem for negatively curved groups, J. Differential Geom. 35 pp 85– (1992) · Zbl 0724.57029
[5] Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 pp 245– (1993) · Zbl 0789.57007
[6] Bowditch, A 4-dimensional Kleinian group, Trans. Amer. Math. Soc. 344 pp 391– (1994) · Zbl 0876.57020
[7] Cooper, Virtually Haken Dehn-filling, J. Differential Geom. 52 pp 173– (1999) · Zbl 1025.57020
[8] Cooper, Some surface subgroups survive surgery, Geom. Topol. 5 pp 347– (2001) · Zbl 1009.57017
[9] Cooper, Essential closed surfaces in bounded 3-manifolds, J. Amer. Math. Soc. 1997 pp 553– (1997) · Zbl 0896.57009
[10] Culler, Dehn surgery on knots, Ann. of Math. 125 ((2)) pp 237– (1987) · Zbl 0633.57006
[11] Culler, Boundary slopes of knots, Comment. Math. Helv. 74 pp 530– (1999) · Zbl 0941.57019
[12] Dahmani, Combination of convergence groups, Geom. Topol 7 pp 933– (2003) · Zbl 1037.20042
[13] Freedman, Haken finiteness for bounded 3-manifolds, locally free groups and cyclic covers, Topology 37 pp 133– (1998) · Zbl 0896.57012
[14] Gitik, On the combination theorem for negatively curved groups, Internat. J. Algebra Comput. 6 pp 751– (1996) · Zbl 0879.20014
[15] Gitik, Graphs and separability properties of groups, J. Algebra 188 pp 125– (1997) · Zbl 0874.20013
[16] Gitik, Ping-pong on negatively curved groups, J. Algebra 217 pp 65– (1999) · Zbl 0936.20019
[17] Hass, Boundary slopes of immersed surfaces in 3-manifolds, J. Differential Geom. 52 pp 303– (1999) · Zbl 0978.57016
[18] Hass, On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds, Proc. Amer. Math. Soc. 130 pp 1851– (2002) · Zbl 0993.57007
[19] Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 pp 373– (1982) · Zbl 0502.57005
[20] Hempel, The finitely generated intersection property for Kleinian groups, Knot theory and manifolds pp 18– · Zbl 0573.20048
[21] Kang, Ideal triangulations of 3-manifolds I: spun normal surface theory, Geom. Topol. Monogr. 7 pp 235– (2004) · Zbl 1085.57016
[22] Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 pp 481– (1987) · Zbl 0596.57011
[23] Maher, Virtually embedded boundary slopes, Topology Appl. 95 pp 63– (1999) · Zbl 0940.57024
[24] Malcev, On isomorphic matrix representations of infinite groups, Mat. Sb. 8 pp 405– (1940) · JFM 66.0088.03
[25] Maskit, On Klein’s combination theorem. IV, Trans. Amer. Math. Soc. 336 pp 265– (1993) · Zbl 0777.30028
[26] Masters, Thick surfaces in hyperbolic 3-manifolds, Geom. Dedicata 119 pp 17– (2006) · Zbl 1095.57018
[27] K. Matsuzaki M. Taniguchi Hyperbolic manifolds and Kleinian groups 1998 Oxford Oxford University Press · Zbl 0892.30035
[28] Oertel, Boundaries of {\(\pi\)}1-injective surfaces, Topology Appl. 78 pp 215– (1997) · Zbl 0879.57014
[29] Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. 2 ((17)) pp 555– (1978) · Zbl 0412.57006
[30] J. P. Serre Trees 2003 Berlin Springer
[31] Susskind, Kleinian groups with intersecting limit sets, J. Anal. Math. 52 pp 26– (1989) · Zbl 0677.30028
[32] Susskind, An infinitely generated intersection of geometrically finite hyperbolic groups, Proc. Amer. Math. Soc 129 pp 2643– (2001) · Zbl 0968.30022
[33] Susskind, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 pp 233– (1992) · Zbl 0791.30039
[34] Thurston, The geometry and topology of three-manifolds, in: Available at (1977)
[35] W. P. Thurston Silvio Levy Three-dimensional geometry and topology 1997 Princeton, NJ Princeton University Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.