By Baker M., Cooper D.
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Additional resources for A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds
M. D. Baker and D. Cooper, ‘Immersed, virtually-embedded, boundary slopes’, Topology Appl. 102 (2000) 239–252. 4. M. Bestvina and M. Feighn, ‘A combination theorem for negatively curved groups’, J. Diﬀerential Geom. 35 (1992) 85–101; Addendum and correction, J. Diﬀerential Geom. 43 (1996) 783–788. 5. B. H. Bowditch, ‘Geometrical ﬁniteness for hyperbolic groups’, J. Funct. Anal. 113 (1993) 245–317 6. B. H. Bowditch and G. Mess, ‘A 4-dimensional Kleinian group’, Trans. Amer. Math. Soc. 344 (1994) 391–405.
The hypothesis of the convex combination theorem that one can thicken without bumping means that there are restrictions on the cusps of the manifolds to be glued. In particular, two rank-1 cusps in the same rank-2 cusp of a 3-manifold cannot be thickened without bumping unless they are parallel. Here is an algebraic viewpoint. Rank-1 cusps give Z subgroups, and two non-parallel cusps in the same rank-2 cusp will generate a Z ⊕ Z group in the group, G, generated by the two subgroups. Thus G is not an amalgamated free product of the subgroups.
Oertel, ‘Boundaries of π 1 -injective surfaces’ Topology Appl. 78 (1997) 215–234. P. Scott, ‘Subgroups of surface groups are almost geometric’, J. London Math. Soc. (2) 17 (1978) 555–565, Correction, J. London Math. Soc. (2) 32 (1985) 217–220. J. P. Serre, Trees (Springer, Berlin, 2003). P. Susskind, ‘Kleinian groups with intersecting limit sets’, J. Anal. Math. 52 (1989) 26–38. P. Susskind, ‘An inﬁnitely generated intersection of geometrically ﬁnite hyperbolic groups’, Proc. Amer. Math. Soc. 129 (2001) 2643–2646.
A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds by Baker M., Cooper D.