Algebraic & Geometric Topology

Convex cocompactness and stability in mapping class groups

Matthew Durham and Samuel J Taylor

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 introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock and is related to questions asked by Farb and Mosher and by Farb.

Article information

Algebr. Geom. Topol., Volume 15, Number 5 (2015), 2837-2857.

Received: 2 July 2014
Revised: 1 January 2015
Accepted: 12 March 2015
First available in Project Euclid: 16 November 2017

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] 51H05: General theory
Secondary: 57M07: Topological methods in group theory 30F60: Teichmüller theory [See also 32G15]

convex cocompact subgroups of mapping class groups stability quasiconvexity hyperbolic groups


Durham, Matthew; Taylor, Samuel J. Convex cocompactness and stability in mapping class groups. Algebr. Geom. Topol. 15 (2015), no. 5, 2837--2857. doi:10.2140/agt.2015.15.2839.

Export citation


  • J,A Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006) 1523–1578
  • J,A Behrstock, Y,N Minsky, Dimension and rank for mapping class groups, Ann. of Math. 167 (2008) 1055–1077
  • M,R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • J Brock, H Masur, Y Minsky, Asymptotics of Weil–Petersson geodesics, II: Bounded geometry and unbounded entropy, Geom. Funct. Anal. 21 (2011) 820–850
  • S Dowdall, R,P Kent, IV, C,J Leininger, Pseudo-Anosov subgroups of fibered 3–manifold groups, Groups Geom. Dyn. 8 (2014) 1247–1282
  • C Dru\commaaccenttu, S Mozes, M Sapir, Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Amer. Math. Soc. 362 (2010) 2451–2505
  • M Duchin, K Rafi, Divergence of geodesics in Teichmüller space and the mapping class group, Geom. Funct. Anal. 19 (2009) 722–742
  • B Farb, Some problems on mapping class groups and moduli space, from: “Problems on mapping class groups and related topics”, (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 11–55
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • B Farb, L Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002) 91–152
  • U Hamenstädt, Word hyperbolic extensions of surface groups, preprint
  • W,J Harvey, Boundary structure of the modular group, from: “Riemann surfaces and related topics”, (I Kra, B Maskit, editors), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245–251
  • R,P Kent, IV, C,J Leininger, Shadows of mapping class groups: capturing convex cocompactness, Geom. Funct. Anal. 18 (2008) 1270–1325
  • R,P Kent, IV, C,J Leininger, S Schleimer, Trees and mapping class groups, J. Reine Angew. Math. 637 (2009) 1–21
  • J Mangahas, S,J Taylor, Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups, preprint
  • H,A Masur, Y,N Minsky, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • H,A Masur, Y,N Minsky, Geometry of the complex of curves, II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902–974
  • H,A Masur, Y,N Minsky, Unstable quasi-geodesics in Teichmüller space, from: “In the tradition of Ahlfors and Bers”, (I Kra, B Maskit, editors), Contemp. Math. 256, Amer. Math. Soc. (2000) 239–241
  • H Min, Hyperbolic graphs of surface groups, Algebr. Geom. Topol. 11 (2011) 449–476
  • Y Minsky, The classification of Kleinian surface groups, I: Models and bounds, Ann. of Math. 171 (2010) 1–107
  • K Rafi, Hyperbolicity in Teichmüller space, Geom. Topol. 18 (2014) 3025–3053
  • K Rafi, S Schleimer, Covers and the curve complex, Geom. Topol. 13 (2009) 2141–2162