Mathematics 210A

Topics in Geometry

Fall 2006
Meeting Mondays, Wednesdays, and Fridays, 2:10-3:00, in 1134 Bainer Hall

Instructor: Lucas Sabalka
Office: 2228 MSB
Office Hours: MT 3:10-4:00, or by appointment

E-mail: sabalka, at
Office Phone: (530) 752-0878

  1. Joseph Gallian. Contemporary abstract algebra, fourth edition. 1998.
  2. Pierre de la Harpe. Topics in geometric group theory. 2000.
  3. Davis B.A. Epstein, James W. Cannon, Derek F. Holt, Silvio V.F. Levy, Michael S. Paterson, and WIlliam P. Thurston. Word processing in groups. 1992.
  4. James Humphreys. Reflection groups and Coxeter groups. 1990.
  5. Ken'ichi Ohshika. Discrete groups. 1998.
  6. John Stallings. Topology of finite graphs. In Inventiones mathematicae 71, 551-565. 1983.
  7. Ruth Charney. An introduction to right-angled Artin groups. 2006.
  8. Joan S. Birman and Tara E. Brendle. Braids: a survey. In Handbook of knot theory, 19-103. 2005.
  9. Ki Hyoung Ko, Sang Jin Lee, Jung Hee Cheon, Jae Woo Han, Ju-sung Kang, and Choonsik Park. New public-key cryptosystem using braid groups. In CRYPTO 2000, LNCS 1880, 166-183. 2000.
Course Syllabus: Click here


Day: Material Covered:
September 29Introduction: Groups (Gallian)
October 2Group Presentations (Gallian)
October 4Group Presentations, Examples (Gallian)
October 6Symmetry Groups (Tom presents) (Gallian)
October 9Frieze Groups (Dustin presents) (Gallian)
October 11Crystallographic Groups (Gallian)
October 13Crystallographic Groups (Gallian)
October 16Cayley Graphs (Wang presents) (Gallian)
October 18Aside: Penrose Tilings, and Cayley Graphs (Gallian, de la Harpe)
October 20BS(2,1) and Word Metric (Epstein et. al.; Gallian)
October 23Braid groups (Birman and Brendle)
October 25Braid groups: Garside theory (Birman and Brendle)
October 27Braid groups: Greedy Normal Form (Katia presents) (Birman and Brendle)
October 30Applications of Braid groups: Knots (Birman and Brendle)
November 1Applications of Braid groups: Alexander's Theorem (Birman and Brendle)
November 3Applications of Braid groups: Cryptography (McCartney presents) (Ko et. al.)
November 6Coxeter groups: Reflection groups (Corrine presents) (Wikipedia)
November 8Catch up: Reidemeister's theorem, Markov's theorem, reflection groups, Coxeter groups (Birman and Brendle; Humphreys; de la Harpe)
November 13Coxeter groups: Coxeter graphs and root systems (Humphreys)
November 15Coxeter groups: Finite Coxeter groups, big results (Wikipedia; Humphreys)
November 17Artin groups and Coxeter groups (Charney)
November 22Stallings Foldings (Stephen and Matt R. present) (Stallings)
November 24Stallings Foldings (Stephen and Matt R. present) (Stallings)
November 27Finite State Automata (Matt L. presents) (Wikipedia)
November 29Automatic groups (Andre presents) (Epstein et. al.)
December 1Automatic groups (Epstein et. al., Ohshika)
December 4Group growth (Sonny presents) (de la Harpe)
December 6Dehn problems (Wang presents) (de la Harpe; Bridson and Haefliger)
December 8Dehn problems and topology