Algebraic & Geometric Topology

Algebraic ending laminations and quasiconvexity

Mahan Mj and Kasra Rafi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence

1 H G Q 1

of hyperbolic groups. These laminations arise in different contexts: existence of Cannon–Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on –trees.

We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite-index subgroups of H , the normal subgroup in the exact sequence above.

Article information

Algebr. Geom. Topol., Volume 18, Number 4 (2018), 1883-1916.

Received: 24 October 2015
Revised: 13 December 2017
Accepted: 24 February 2018
First available in Project Euclid: 3 May 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 30F60: Teichmüller theory [See also 32G15]

hyperbolic group mapping torus quasiconvexity ending lamination Cannon-Thurston map


Mj, Mahan; Rafi, Kasra. Algebraic ending laminations and quasiconvexity. Algebr. Geom. Topol. 18 (2018), no. 4, 1883--1916. doi:10.2140/agt.2018.18.1883.

Export citation


  • M Bestvina, $\mathbb R\!$–trees in topology, geometry, and group theory, from “Handbook of geometric topology” (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 55–91
  • M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85–101
  • M Bestvina, M Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014) 104–155
  • M Bestvina, M Feighn, M Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997) 215–244
  • M Bestvina, P Reynolds, The boundary of the complex of free factors, Duke Math. J. 164 (2015) 2213–2251
  • B H Bowditch, The Cannon–Thurston map for punctured-surface groups, Math. Z. 255 (2007) 35–76
  • B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) 1250016, 66
  • J W Cannon, W P Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007) 1315–1355
  • T Coulbois, A Hilion, M Lustig, Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\mathbb R\!$–trees, Illinois J. Math. 51 (2007) 897–911
  • T Coulbois, A Hilion, M Lustig, $\mathbb R\!$–trees and laminations for free groups, I: Algebraic laminations, J. Lond. Math. Soc. 78 (2008) 723–736
  • T Coulbois, A Hilion, M Lustig, $\mathbb R\!$–trees and laminations for free groups, II: The dual lamination of an $\mathbb R\!$–tree, J. Lond. Math. Soc. 78 (2008) 737–754
  • T Coulbois, A Hilion, P Reynolds, Indecomposable $F_{\!N}\!$–trees and minimal laminations, Groups Geom. Dyn. 9 (2015) 567–597
  • M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91–119
  • S Dowdall, I Kapovich, S J Taylor, Cannon–Thurston maps for hyperbolic free group extensions, Israel J. Math. 216 (2016) 753–797
  • S Dowdall, R P Kent, IV, C J Leininger, Pseudo-Anosov subgroups of fibered $3$–manifold groups, Groups Geom. Dyn. 8 (2014) 1247–1282
  • S Dowdall, S J Taylor, The co-surface graph and the geometry of hyperbolic free group extensions, J. Topol. 10 (2017) 447–482
  • S Dowdall, S J Taylor, Hyperbolic extensions of free groups, Geom. Topol. 22 (2018) 517–570
  • B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810–840
  • B Farb, L Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002) 91–152
  • M Gromov, Hyperbolic groups, from “Essays in group theory” (S M Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75–263
  • M Gromov, Asymptotic invariants of infinite groups, from “Geometric group theory, 2” (G A Niblo, M A Roller, editors), London Math. Soc. Lect. Note Ser. 182, Cambridge Univ. Press (1993) 1–295
  • V Guirardel, Actions of finitely generated groups on $\mathbb R\!$–trees, Ann. Inst. Fourier $($Grenoble$)$ 58 (2008) 159–211
  • U Hamenstädt, Word hyperbolic extensions of surface groups, preprint (2005)
  • U Hamenstädt, S Hensel, Stability in outer space, preprint (2014)
  • G C Hruska, D T Wise, Packing subgroups in relatively hyperbolic groups, Geom. Topol. 13 (2009) 1945–1988
  • I Kapovich, M Lustig, Intersection form, laminations and currents on free groups, Geom. Funct. Anal. 19 (2010) 1426–1467
  • I Kapovich, M Lustig, Cannon–Thurston fibers for iwip automorphisms of $F_N$, J. Lond. Math. Soc. 91 (2015) 203–224
  • R P Kent, IV, C J Leininger, Shadows of mapping class groups: capturing convex cocompactness, Geom. Funct. Anal. 18 (2008) 1270–1325
  • E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (1999)
  • H Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982) 183–190
  • H Masur, Y N Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902–974
  • Y N Minsky, Teichmüller geodesics and ends of hyperbolic $3$–manifolds, Topology 32 (1993) 625–647
  • Y N Minsky, On rigidity, limit sets, and end invariants of hyperbolic $3$–manifolds, J. Amer. Math. Soc. 7 (1994) 539–588
  • Y N Minsky, Bounded geometry for Kleinian groups, Invent. Math. 146 (2001) 143–192
  • M Mitra, Ending laminations for hyperbolic group extensions, Geom. Funct. Anal. 7 (1997) 379–402
  • M Mitra, Cannon–Thurston maps for hyperbolic group extensions, Topology 37 (1998) 527–538
  • M Mitra, On a theorem of Scott and Swarup, Proc. Amer. Math. Soc. 127 (1999) 1625–1631
  • M Mj, Cannon–Thurston maps and bounded geometry, from “Teichmüller theory and moduli problems” (I Biswas, R S Kulkarni, S Mitra, editors), Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Math. Soc., Mysore (2010) 489–511
  • M Mj, Ending laminations and Cannon–Thurston maps, Geom. Funct. Anal. 24 (2014) 297–321
  • M Mj, Cannon–Thurston maps for Kleinian groups, Forum Math. Pi 5 (2017) art. id. e1, 49 pages
  • M Mj, A Pal, Relative hyperbolicity, trees of spaces and Cannon–Thurston maps, Geom. Dedicata 151 (2011) 59–78
  • M Mj, P Sardar, A combination theorem for metric bundles, Geom. Funct. Anal. 22 (2012) 1636–1707
  • L Mosher, Hyperbolic extensions of groups, J. Pure Appl. Algebra 110 (1996) 305–314
  • A Pal, Relatively hyperbolic extensions of groups and Cannon–Thurston maps, Proc. Indian Acad. Sci. Math. Sci. 120 (2010) 57–68
  • K Rafi, Hyperbolicity in Teichmüller space, Geom. Topol. 18 (2014) 3025–3053
  • P Reynolds, On indecomposable trees in the boundary of outer space, Geom. Dedicata 153 (2011) 59–71
  • P Reynolds, Reducing systems for very small trees, preprint (2012)
  • G P Scott, G A Swarup, Geometric finiteness of certain Kleinian groups, Proc. Amer. Math. Soc. 109 (1990) 765–768
  • P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. 17 (1978) 555–565
  • W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url {\unhbox0