Non–positively curved groups
Math 7390 — Topics in Algebra

Mondays and Wednesdays, 10:10–11:25, Malott Hall 205
Lecturer: Tim Riley

Synopsis. This course will be an introduction to classes of infinite discrete groups regarded as non–positively curved — in particular, Gromov–hyperbolic, CAT(-1), CAT(0), automatic, semi–hyperbolic, and systolic groups.

Arguably, the subject has three points of origin. The oldest lies with Max Dehn and concerns combinatorial techniques to solve problems in low–dimensional topology. More recently, Misha Gromov carried ideas from Riemannian Geometry into the coarse word of discrete groups and Jim Cannon studied algorithmic and combinatorial aspects of Cayley graphs related to group actions on hyperbolic space.

I will explain some of the implications of non–positive curvature assumptions on groups in contexts such as subgroup structure, boundaries, grammatical complexity of normal forms, the Borel Conjecture, the Baum–Connes Conjecture, and the Novikov Conjecture. I will survey some of the open problems of the subject.

References
  1. M.Bestvina, G.Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991) 469–481.
  2. B.H.Bowditch, A course on Geometric Group Theory
  3. N.Brady, T.Riley and H.Short, The Geometry of the Word Problem for Finitely Generated Groups
  4. M.R.Bridson and A.Haefliger, Metric Spaces of Non-Positive Curvature
  5. J.W. Cannon and W.P.Thurston, Group invariant Peano curves
  6. D.J.Collins, R.I.Grigorchuk, P.F.Kurchanov and H.Zieschang, Combinatorial Group Theory and Applications to Geometry
  7. M.Coornaert, T.Delzant, A.Papadopoulos, Géométrie et théorie des groupes: les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990
  8. M.Dehn, trans. J.Stillwell, Papers on Group Theory and Topology
  9. D.Epstein et al., Word processing in groups
  10. S.M.Gersten, Introduction to hyperbolic and automatic groups
  11. M.Gromov, Asymptotic Invariants of Infinte Groups
  12. M.Gromov, Hyperbolic groups, Essays in Group Theory, S. Gersten ed., MSRI Publications 8 (1987), 75–265, Springer
  13. É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics, 83, Birkhauser Boston, Inc., Boston, MA, 1990, xii+285 pp.
  14. I.Kapovich and N.Benakli, Boundaries of hyperbolic groups
  15. W.Lueck, Survey on geometric group theory
  16. R.C.Lyndon and P.E.Schupp, Combinatorial Group Theory
  17. M.Mitra, Cannon–Thurston maps for hyperbolic group extensions, Topology, vol. 37 (1998), no. 3, pp. 527–538
  18. M.Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces, Journal of Differential Geometry, vol. 48 (1998), no. 1, pp. 135–164
  19. H.Short et al., Notes on word hyperbolic groups, Group Theory from a Geometrical Viewpoint (E.Ghys, A.Haefliger, A.Verjovsky, ed) Proc. ICTP Trieste 1990, WorldScientific, Singapore, 1991, 3–64
  20. The World of Groups — open problems list

Tim Riley