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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)
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##### 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
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