# zbMATH — the first resource for mathematics

Cannon-Thurston maps for pared manifolds of bounded geometry. (English) Zbl 1166.57009
The author uses the following terminology:
A pared manifold is a pair $$(M,P)$$ where $$M$$ is a $$3$$-manifold with boundary and $$P$$ is a (possibly empty) $$2$$-dimensional submanifold with boundary of $$\partial M$$ such that:
1)
the fundamental group of each component of $$P$$ injects into the fundamental group of $$M$$ and it contains an abelian subgroup of finite index;
2)
any cylinder $$C:(S^1\times I,\partial(S^1\times I))\to (M,P)$$ with $$C_*:\pi_1(S^1\times I)\to \pi_1(M)$$ injective is homotopic rel. boundary to $$P$$;
3)
$$P$$ contains every component of $$\partial M$$ which has an abelian subgroup of finite index.
The definition is due to Thurston.
A pared manifold $$(M,P)$$ is said to have incompressible boundary if each component of $$M\setminus P$$ is incompressible in $$M$$.
A manifold $$M$$ equipped with a metric is said to have bounded geometry if in a complement of the cusps the injectivity radius is bounded below by some positive number.

The main result of the paper is the following
Theorem: Let $$N^h$$ be a hyperbolic structure of bounded geometry on a pared manifold $$(M,P)$$ with incompressible boundary $$\partial_0M=(\partial M-P)$$ and let $$M_{gf}$$ be a geometrically finite hyperbolic structure adapted to $$(M,P)$$. Then the map $$i:\widetilde{M_{gf}}\to \widetilde{N^h}$$ between the universal covers extends continuously to a map between the Gromov boundaries $$\widehat{M_{gf}}\to\widehat{N^h}$$. Furthermore, the limit set of $$\widetilde{M}$$ is locally connected.
This extension map between the boundaries is an example of a Cannon-Thurston map. The result of this paper is one of several results by the author on similar problems of existence of Cannon-Thurston maps, written in a series of papers.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 20F67 Hyperbolic groups and nonpositively curved groups 57N16 Geometric structures on manifolds of high or arbitrary dimension
Full Text:
##### References:
 [1] W Abikoff, Kleinian groups-geometrically finite and geometrically perverse, Contemp. Math. 74, Amer. Math. Soc. (1988) 1 · Zbl 0662.30043 [2] J W Anderson, B Maskit, On the local connectivity of limit set of Kleinian groups, Complex Variables Theory Appl. 31 (1996) 177 · Zbl 0869.30034 [3] M Bestvina, Geometric group theory problem list (2004) · math.utah.edu [4] M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85 · Zbl 0724.57029 · euclid:jdg/1214447806 [5] M Bestvina, M Feighn, M Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997) 215 · Zbl 0884.57002 · doi:10.1007/PL00001618 [6] M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. $$(2)$$ 135 (1992) 1 · Zbl 0757.57004 · doi:10.2307/2946562 [7] F Bonahon, Bouts des variétés hyperboliques de dimension $$3$$, Ann. of Math. $$(2)$$ 124 (1986) 71 · Zbl 0671.57008 · doi:10.2307/1971388 [8] B H Bowditch, Relatively hyperbolic groups, preprint, Southampton (1997) · Zbl 1259.20052 [9] B H Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998) 643 · Zbl 0906.20022 · doi:10.1090/S0894-0347-98-00264-1 [10] B H Bowditch, Convergence groups and configuration spaces (editors J Cossey, C F Miller, W D Neumann, M Shapiro), de Gruyter (1999) 23 · Zbl 0952.20032 [11] B H Bowditch, Stacks of hyperbolic spaces and ends of 3 manifolds, preprint, Southampton (2002) · Zbl 1297.57044 [12] B H Bowditch, The Cannon-Thurston map for punctured-surface groups, Math. Z. 255 (2007) 35 · Zbl 1138.57020 · doi:10.1007/s00209-006-0012-4 [13] J F Brock, Iteration of mapping classes and limits of hyperbolic 3-manifolds, Invent. Math. 143 (2001) 523 · Zbl 0969.57011 · doi:10.1007/s002220000114 [14] J F Brock, R D Canary, Y N Minsky, The Classification of Kleinian surface groups II: The Ending Lamination Conjecture, preprint (2004) · Zbl 1253.57009 · doi:10.4007/annals.2012.176.1.1 [15] J W Cannon, W P Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007) 1315 · Zbl 1136.57009 · doi:10.2140/gt.2007.11.1315 [16] M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics 1441, Springer (1990) · Zbl 0727.20018 · doi:10.1007/BFb0084913 [17] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027 · doi:10.1007/s000390050075 [18] W J Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980) 205 · Zbl 0428.20022 · doi:10.1007/BF01418926 · eudml:142707 [19] É Ghys, P d l Harpe, editors, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser (1990) · Zbl 0731.20025 [20] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 [21] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser (1999) · Zbl 0953.53002 [22] J G Hocking, G S Young, Topology, Addison-Wesley Publishing Co., Reading, MA-London (1961) · Zbl 0135.22701 [23] E Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere, Amer. J. Math. 121 (1999) 1031 · Zbl 1011.30035 · doi:10.1353/ajm.1999.0034 · muse.jhu.edu [24] C T McMullen, Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989) 95 · Zbl 0672.30017 · doi:10.1007/BF01850656 · eudml:143694 [25] C T McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990) 425 · Zbl 0695.57012 · doi:10.1007/BF01234427 · eudml:143766 [26] C T McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001) 35 · Zbl 1061.37025 · doi:10.1007/PL00005809 [27] Y N Minsky, Teichmüller geodesics and ends of hyperbolic $$3$$-manifolds, Topology 32 (1993) 625 · Zbl 0793.58010 · doi:10.1016/0040-9383(93)90013-L [28] Y N Minsky, On rigidity, limit sets, and end invariants of hyperbolic $$3$$-manifolds, J. Amer. Math. Soc. 7 (1994) 539 · Zbl 0808.30027 · doi:10.2307/2152785 [29] Y N Minsky, The classification of punctured-torus groups, Ann. of Math. $$(2)$$ 149 (1999) 559 · Zbl 0939.30034 · doi:10.2307/120976 · www.math.princeton.edu · eudml:120365 [30] Y N Minsky, Bounded geometry for Kleinian groups, Invent. Math. 146 (2001) 143 · Zbl 1061.37026 · doi:10.1007/s002220100163 · arxiv:math/0105078 [31] Y N Minsky, The classification of Kleinian surface groups I: Models and Bounds, preprint (2002) · Zbl 1193.30063 · doi:10.4007/annals.2010.171.1 · annals.princeton.edu [32] Y N Minsky, End invariants and the classification of hyperbolic 3-manifolds, Int. Press, Somerville, MA (2003) 181 · Zbl 1049.57010 [33] M Mitra, PhD thesis, UC Berkeley (1997) [34] M Mitra, Ending laminations for hyperbolic group extensions, Geom. Funct. Anal. 7 (1997) 379 · Zbl 0880.57001 · doi:10.1007/PL00001624 [35] M Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology 37 (1998) 527 · Zbl 0907.20038 · doi:10.1016/S0040-9383(97)00036-0 [36] M Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998) 135 · Zbl 0906.20023 · euclid:jdg/1214460609 [37] H Miyachi, Semiconjugacies between actions of topologically tame Kleinian groups, preprint (2002) [38] M Mj, Cannon-Thurston Maps for Surface Groups I: Amalgamation Geometry and Split Geometry (2005) · arxiv:math.GT/0512539 [39] M Mj, Cannon-Thurston maps, i-bounded geometry and a theorem of McMullen (2005) · Zbl 1237.57018 · arxiv:math.GT/0511041 · eudml:116466 [40] M Mj, Cannon-Thurston Maps for Surface Groups II: Split Geometry and the Minsky Model (2006) · arxiv:math.GT0607509 [41] M Mj, A Pal, Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps (2007) · Zbl 1222.57013 · doi:10.1007/s10711-010-9519-2 · arxiv:0708.3578 [42] R M Myers, From Beowulf to Virginia Woolf: An astounding and wholly unauthorized history of English literature, Bobbs-Merrill, Indianapolis (1952) [43] A Pal, Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps (2008) · arxiv:0801.0933 [44] G P Scott, Compact submanifolds of $$3$$-manifolds, J. London Math. Soc. $$(2)$$ 7 (1973) 246 · Zbl 0266.57001 · doi:10.1112/jlms/s2-7.2.246 [45] J Souto, Cannon-Thurston maps for thick free groups, preprint (2006) [46] W P Thurston, The geometry and topology of 3-manifolds, Princeton University notes (1980) [47] W P Thurston, Hyperbolic structures on $$3$$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. $$(2)$$ 124 (1986) 203 · Zbl 0668.57015 · doi:10.2307/1971277 [48] W P Thurston, Hyperbolic structures on $$3$$-manifolds. III. Deformation of 3-manifolds with incompressible boundary, preprint (1986) · Zbl 0668.57015 · doi:10.2307/1971277 [49] A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41 · Zbl 1043.20020 · doi:10.1515/crll.2004.007
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.