Geometry & Topology

One-ended subgroups of graphs of free groups with cyclic edge groups

Henry Wilton

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Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. As a corollary, the same holds for limit groups. We also obtain a characterisation of surfaces with boundary among free groups equipped with peripheral structures.

Article information

Geom. Topol., Volume 16, Number 2 (2012), 665-683.

Received: 28 February 2011
Revised: 1 March 2011
Accepted: 6 January 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups 57M07: Topological methods in group theory

free group hyperbolic group surface subgroup


Wilton, Henry. One-ended subgroups of graphs of free groups with cyclic edge groups. Geom. Topol. 16 (2012), no. 2, 665--683. doi:10.2140/gt.2012.16.665.

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  • E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459–466
  • H Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993) 3–47
  • J Berge, Heegaard documentation (2010) Available at \setbox0\makeatletter\@url {\unhbox0
  • M Bestvina, Questions in geometric group theory, preprint (2004) Available at \setbox0\makeatletter\@url {\unhbox0
  • M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85–101
  • B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145–186
  • M R Bridson, Problems concerning hyperbolic and $\mathrm{CAT}(0)$ groups, preprint (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • D Calegari, Surface subgroups from homology, Geom. Topol. 12 (2008) 1995–2007
  • C H Cashen, Splitting line patterns in free groups
  • C H Cashen, N Macura, Line patterns in free groups, Geom. Topol. 15 (2011) 1419–1475
  • F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933–963
  • G-A Diao, M Feighn, The Grushko decomposition of a finite graph of finite rank free groups: an algorithm, Geom. Topol. 9 (2005) 1835–1880
  • C Gordon, H Wilton, On surface subgroups of doubles of free groups, J. Lond. Math. Soc. 82 (2010) 17–31
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, (S M Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
  • V Guirardel, G Levitt, JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space
  • F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551–1620
  • M Hall, Jr, Subgroups of finite index in free groups, Canadian J. Math. 1 (1949) 187–190
  • T Hsu, D T Wise, Cubulating graphs of free groups with cyclic edge groups, Amer. J. Math. 132 (2010) 1153–1188
  • J Kahn, V Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, to appear in Ann. of Math.
  • O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998) 472–516
  • S-h Kim, On right-angled Artin groups without surface subgroups, Groups Geom. Dyn. 4 (2010) 275–307
  • S-h Kim, H Wilton, Polygonal words in free groups, to appear in Q. J. Math.
  • L Louder, Scott complexity and adjoining roots to finitely generated groups, to appear in Groups Geom. Dyn.
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergeb. Math. Grenz. 89, Springer, Berlin (1977)
  • J F Manning, Virtually geometric words and Whitehead's algorithm, Math. Res. Lett. 17 (2010) 917–925
  • P Scott, Finitely generated $3$–manifold groups are finitely presented, J. London Math. Soc. 6 (1973) 437–440
  • P Scott, T Wall, Topological methods in group theory, from: “Homological group theory (Proc. Sympos., Durham, 1977)”, (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137–203
  • Z Sela, Diophantine geometry over groups I: Makanin–Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31–105
  • A Shenitzer, Decomposition of a group with a single defining relation into a free product, Proc. Amer. Math. Soc. 6 (1955) 273–279
  • H Wilton, Elementarily free groups are subgroup separable, Proc. Lond. Math. Soc. 95 (2007) 473–496
  • H Wilton, Hall's theorem for limit groups, Geom. Funct. Anal. 18 (2008) 271–303
  • D T Wise, Subgroup separability of graphs of free groups with cyclic edge groups, Q. J. Math. 51 (2000) 107–129