WORD-HYPERBOLIC GROUPS
FALL 2001  MATH 415 Section D1
MWF, 11am    Altgeld Hall, rm 441

WWW:     http://www.math.uiuc.edu/~kapovich/415-01/415-01.html





Telephone: 265-0633
e-mail: kapovich@math.uiuc.edu. (Preferred method of reaching me!)
Office location: Illini Hall, room 328
Office hours:   Tuesday, Friday, 3-4pm  AND by appointment
 

TEXTBOOK:  There is no assigned textbook for this course. We will be using several articles as our primary sources. All of those will be directly supplied by the instructor to the course  participants.
 
 

Here are some of the sources that we will use for of the course (we shall start with those in red):
 
 

  1. "Notes on word-hyperbolic groups", by J.M.Alonso, T.Brady, D.Cooper, V.Ferlini, M.Lustig, M.Mihalik, M.Shapiro, H.Short; in Group Theory from a Geometric Viewpoint, World Scientific, 1991, pp. 3-63
  2. "The Theory of negatively curved spaces and groups",  by J.Cannon; in "Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces", Oxford University Press, 1991,  pp. 315-368
  3. "Metric Spaces of Non-Positive Curvature", by M.Bridson and A.Haefliger; Springer-Verlag, 1999 (selected chapters)
  4. "Greenberg's theorem for quasiconvex subgroups of word hyperbolic groups", by I.Kapovich and H.Short; Canadiand J. Math 48 (1996), no. 6, pp. 1224-1244
  5. "On Hausdorff-Gromov convergence and a theorem of Paulin", by M.Bridson and G.Swarup;  Enseign. Math. (2) 40 (1994), no. 3-4,  pp. 267-28
  6. "Hyperbolic groups", by M.Gromov; Essays in group theory, 75--263,  Math. Sci. Res. Inst. Publ., 8,  Springer, New York, 1987 [this monograph is recommended ONLY for reference purposes!]
  7. "Cut points and canonical splittings of hyperbolic groups", by B.Bowditch; Acta Math. 180 (1998), no. 2,  pp. 145-186
  8. "Cannon-Thurston maps for hyperbolic group extensions", by M.Mitra;  Topology 37 (1998), no. 3,  pp. 527-538
  9. "Word Processing in Groups", by D.Epstein,  J.Cannon,  D.Holt,  S.Levy,  M.Paterson, and W.Thurston; Jones and Bartlett Publishers, Boston, MA, 1992 (selected topics)
  10. "Finite subgroups of hyperbolic groups", by N.Brady; Internat. J. Algebra Comput. 10 (2000), no. 4,  pp. 399-405
  11. "A combination theorem for negatively curved groups", by M.Bestvina and M.Feighn; J. Differential Geom. 35 (1992), no. 1, pp. 85-101
  12. "The fixed group of an automorphism of a word hyperbolic group is rational",  by W.Neumann; Invent. Math. 110 (1992), no. 1, pp. 147-150
  13. "Almost every group is hyperbolic",  by A.Yu. Ol'shanskii; Internat. J. Algebra Comput. 2 (1992), no. 1,  pp. 1-17
  14.  "Combinatorial Group Theory",  by R.Lyndon and P.Schupp; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. Springer-Verlag,

    15. Berlin-New York, 1977


                                                                              Syllabus (very approximate):

  1. Introduction to Geometric Group Theory. Cayley graphs, Quasi-isometries and Geometric Actions.
  2. Gromov-Hyperbolic metric spaces and word-hyperbolic groups.
  3. Divergence of geodesics. Quasigeodesics and quasiconvex subsets in hyperbolic spaces.
  4. Isoperimetric functions and Dehn algorithm.
  5. Hyperbolic groups are automatic.
  6. Hyperbolicity and Quasi-isometries. Equivalence of various definitions of hyperbolicity. Sources of hyperbolic groups and hyperbolicity as a generic property.
  7. Conjugacy Problem in Hyperbolic groups.
  8. Rips Complex and basic cohomological properties of hyperbolioc groups.
  9. Finite subgroups of hyperbolic groups.
  10. Hyperbolic boundary: topological and metric structure.
  11. Basic subgroup properties of hyperbolic groups.
  12. Action on the boundary as a convergence action.
  13. Limit sets and quasiconvexity of subgroups.
  14. Free and non-free subgroups of hyperbolic groups. Rips' construction.
  15. Amalgamated free products, HNN extensions and the Combination Theorem.
  16. Automorphisms of Hyperbolic groups. Neumann's Theorem on the fixed subgroup of an automorphism.
  17. Outer automorphisms, Dehn twists and small actions on R-trees.
  18. Cannon-Thurston map.
  19. Bowditch's approach to the JSJ decomposition for hyperbolic groups.
  20. Other topics, time permitting

 
 
 
 
 

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