Sources by topic

1. Background — Bridson universe of finitely presented groups — Bridson's 2006 ICM article

   • Bridson, On the subgroups of semihyperbolic groups

   • McCammond, Constructing non-positively curved spaces and groups

   • Bridson and Haefliger, Metric Spaces of Non-Positive Curvature

2. 
   Rips' Construction

   • Rips, Subgroups of small cancellation groups, Bull. London Math Soc., 14(1):45–47, 1982

   • Wise's residually finite version

   • Section 5.1 in Bridson's 2006 ICM article

   • Application to constructing hyperbolic groups with subgroups of distortion not bounded above by a recursive function (see Gromov's Asymptotic Invariants of Infinite Discrete Groups, Pittet's thesis and Arzhantseva–Osin)

   • Baumslag, Miller, and Short, Unsolvable problems about small cancellation and word hyperbolic groups

3. 
   Subgroups of Fn × Fn

   • Mihailova's Construction

     • K. A. Mihailova, The occurrence problem for direct products of groups, Dokl. Acad. Nauk SSRR 119 (1958), 1103-1105

     • Bogopolski and Ventura, A recursive presentation for Mihailova's subgroup

     • Application to relating subgroup distortion in F2 × F2 to the Dehn functions of finitely presented groups: Theorem 2 in Olshaknii and Sapir's, Length and area functions on groups and quasi-isometric Higman embeddings

   • Finitely presented subgroups: see references to Grunewald, Baumslag–Roseblade, Short, Bridson–Wise in the introduction to Bogopolski–Ventura

4. 
   Finiteness properties and subgroups of Fn × ··· × Fn

   • Stallings' group

     • Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. J. Math., 85:541–543, 1963

     • Section 4.4 of Murray Elder's thesis includes a good summary of different presentations for Stallings' group

   • Beiri–Stallings groups

     • R. Bieri. Homological dimension of discrete groups. Queen Mary Lecture Notes, 1976

     • Section 2.4 of N. Brady, Dehn Functions and Non-Positive Curvature, CRM Notes

     • Appendix D of Kai-Uwe Bux, Groups and Space, Volume 1: Groups

   • Brown, Cohomology of Groups

5. 
   Bestvina–Brady Groups — subgroups of right–angled Artin groups

   • Bestvina and N. Brady

   • Kai-Uwe Bux's account

   • N. Brady, Dehn Functions and Non-Positive Curvature, CRM Notes

6. 
   Distortion of finitely presented (or, better, finite–rank free) subgroups in CAT(0) or hyperbolic groups

   • Z–subgroups are undistorted

   • Dison and Riley's Hydra groups — CAT(0) groups with hugely distorted free subgroups

   • Barnard, Brady & Dani's, Super-exponential distortion of subgroups of CAT(-1) groups

   • Mitra's examples: in Section 5 of Cannon-Thurston maps for trees of hyperbolic metric spaces

   • Dison, N. Brady and Riley's hyperbolic hydra — hyperbolic groups with hugely distorted free subgroups

   • Baumslag, Bridson, Miller, and Short, Fibre products, non-positive curvature, and decision problems — groups G that are both CAT(0) and hyperbolic and yet such that G × G has a finitely presented subgroup whose distortion is not bounded above by any recursive function. See also the 1-2-3 Theorem: 5.16 in III.Gamma of Bridson and Haefliger, Metric Spaces of Non-Positive Curvature.

7. 
   Dehn functions of subgroups of CAT(0) groups

   • Bridson's exponential example: end of chapter 6 in Part III.Gamma of Bridson and Haefliger, Metric Spaces of Non-Positive Curvature

   • Section 2 of N. Brady, Dehn Functions and Non-Positive Curvature, CRM Notes

   • Dison's thesis, J. Group Theory, and Bull. Lond. Math. Soc. articles

   • Abrams, N. Brady, Dani, Duchin, and Young, Pushing fillings in right-angled Artin groups

   • N. Brady, D. Guralnik, S. R. Lee, Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups

8. 
   Subdirect products of surface groups or limit groups or hyperbolic groups

   • Bridson and Miller, Recognition of subgroups of direct products of hyperbolic groups — the world of finitely presented subgroups of the product of three hyperbolic groups is shown to be wild

   • Bridson, Howie, Miller, Short, Subgroups of direct products of limit groups

   • Bridson, Howie, Miller, Short, Finitely presented residually free groups

   • Bridson, Howie, Miller, Short, Subgroups of direct products of surface groups

9. 
   The Tits alternative for hyperbolic groups and for CAT(0) groups

   • The Tits Alternative for hyperbolic groups is proved in Gromov, Hyperbolic groups, In Essays in group theory, vol. 8 of Math. Sci. Res. Inst. Publ., pages 75–263, Springer, 1987

   • The Tits Alternative for CAT(0) groups is open

     • Section 7 of Bridson, Problems concerning hyperbolic and CAT(0) groups

     • Sageev and Wise, The Tits Alternative for CAT(0) Cubical Complexes

     • Xie, Groups Acting on CAT(0) Square Complexes

     • Question about the existence of fake planes — Problem 1.9 in McCammond's survey and Question 2.1 in Bridson's problem list

10. 
    Finitely presented subgroups of hyperbolic groups

    • Gersten: if a hyperbolic group has cohomological dimension 2 then all its finitely presnted subgroups are hyperbolic

    • Brady's example of a non–hyperbolic finitely presented subgroup of a hyperbolic group

    • Gersten, Some Remarks On Subgroups Of Hyperbolic Groups

    • Section 4 of Bridson, Problems concerning hyperbolic and CAT(0) groups

    • Section 2.6 of N. Brady, Dehn Functions and Non-Positive Curvature, CRM Notes

    • N. Brady, Clay, Dani Morse theory and conjugacy classes of finite subgrousp II

11. 
    

12. Cannon–Thurston Maps

    • Cannon and Thurston, Group invariant Peano curves

    • Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces

    • I. Kapovich & N. Benakli, Boundaries of hyperbolic groups

    • M. Kapovich, Boundaries (Problems in Geometry Group Theory)

13. 
    Question (Gromov). Does every 1-ended word-hyperbolic group contain a hyperbolic surface group?

    • The virtually Haken Conjecture

    • Section 4 of Bridson, Problems concerning hyperbolic and CAT(0) groups

    • Gordon, Long and Reid, Surface subgroups of Coxeter and Artin groups

    • Calegari, Surface subgroups from homology

    • Kahn and Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold

    • Gordon and Wilton, On surface subgroups of doubles of free group

    • Sang–hyun (Sam) Kim's papers

    • Wilton, One-ended subgroups of graphs of free groups with cyclic edge groups

Textbooks / General References

1. N.Brady, Riley, Short, The Geometry of the Word Problem for Finitely Generated Groups

2. Bridson and Haefliger, Metric Spaces of Non-Positive Curvature

3. Brown, Cohomology of Groups

4. Bux, Groups and Spaces, Volume I: Groups, Volume II: Spaces

5. Davis, The Geometry and Topology of Coxeter Group

6. Epstein et al., Word Processing in Groups

7. Geoghegan, Topological Methods in Group Theory Theory

8. Gromov, Asymptotic Invariants of Infinite Discrete Groups

9. de la Harpe, Topics in Geometric Group Theory

10. Lyndon and Schupp, Combinatorial Group Theory

11. Meier, Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups

Lists of open questions

1. Problems in Geometric Group Theory Wiki

2. Baumslag, Myasnikov and Shpilrain, Open Problems in combinatorial and geometric group theory

3. Open Problem Garden

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Resources on blogging mathematics

• WordPress.com

• WordPress' support pages on including LaTeX in their blogs

• Advice from Henry Wilton on blogging LaTeX

• LaTeX2WP: a program by Luca Trevisan that converts LaTeX into something that can be cut and pasted into WordPress

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  • Using LaTeX2WP

• John Baez, AMS opinion piece on Math Blogs and an associated blog page