Sources by topic 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 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 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 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 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 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. 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 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 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 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 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) 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 N.Brady, Riley, Short, The Geometry of the Word Problem for Finitely Generated Groups Bridson and Haefliger, Metric Spaces of Non-Positive Curvature Brown, Cohomology of Groups Bux, Groups and Spaces, Volume I: Groups, Volume II: Spaces Davis, The Geometry and Topology of Coxeter Group Epstein et al., Word Processing in Groups Geoghegan, Topological Methods in Group Theory Theory Gromov, Asymptotic Invariants of Infinite Discrete Groups de la Harpe, Topics in Geometric Group Theory Lyndon and Schupp, Combinatorial Group Theory Meier, Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups Lists of open questions Problems in Geometric Group Theory Wiki Baumslag, Myasnikov and Shpilrain, Open Problems in combinatorial and geometric group theory Open Problem Garden