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
Dedicata 119 (2006) 17–33. K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups (Oxford University Press, Oxford, 1998). U. 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.
It follows that there exist a ˜ i that has a tubular neighbourhood of radius κ. The ˜ i of Mi and a lift of Pi to M ﬁnite cover M result follows from the previous theorem. 7. Subgroup separability Two groups G1 and G2 are commensurable if there is a group H that is isomorphic to ﬁnite index subgroups of both G1 and G2 . Two path-connected topological spaces X, Y ˜ Y˜ that are homeomorphic. Clearly, are commensurable if there are ﬁnite-sheeted covers X, commensurable spaces have commensurable fundamental groups.
Culler, C. McA. Gordon, J. Luecke and P. B. Shalen, ‘Dehn surgery on knots’, Ann. of Math. (2) 125 (1987) 237–300. 11. M. Culler and P. B. Shalen, ‘Boundary slopes of knots’, Comment. Math. Helv. 74 (1999) 530–547. 12. F. ’ Geom. Topol. 7 (2003) 933–963. 13. B. Freedman and M. H. Freedman, ‘Haken ﬁniteness for bounded 3-manifolds, locally free groups and cyclic covers’, Topology 37 (1998) 133–147. 14. R. Gitik, ‘On the combination theorem for negatively curved groups’, Internat. J. Algebra Comput.
A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds by Baker M., Cooper D.