401-3203-67L  Small Cancellation Theory

SemesterHerbstsemester 2017
DozierendeD. Gruber
Periodizitäteinmalige Veranstaltung
LehrspracheEnglisch


KurzbeschreibungSmall cancellation theory studies groups given by presentations in which defining relations have small common subwords. By translating group theoretic questions into geometric objects and applying concepts of negative curvature, it produces a variety of theorems on infinite groups. We will give an introduction to the theory, discuss important results, and touch on more recent developments.
LernzielFamiliarity with the fundamental methods of small cancellation theory and its main applications; ability to apply the methods to create new examples of infinite groups with prescribed properties; basic understanding of connections with Gromov hyperbolicity.
InhaltWe plan to cover a selection (depending on time) of the following topics:
- Methods of classical small cancellation theory (e.g. van Kampen diagrams, van Kampen's lemma, Greendlinger's lemma)
- Fundamental properties of small cancellation groups (e.g. Torsion Theorem, asphericity, linear/quadratic Dehn function)
- Connections with algorithmic decision problems in groups (e.g. Dehn's algorithm for solving the word problem in surface groups, solvability of word and conjugacy problems in small cancellation groups)
- Easy examples of small cancellation monsters (e.g. Pride's example, Rips construction)
- Graphical generalization of small cancellation theory and applications (e.g. groups with expander graphs embedded in their Cayley graphs)
- Connections with Gromov hyperbolicity
LiteraturV. Guirardel, Geometric small cancellation. Geometric group theory, 55-90, IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, RI, 2014.

R. C. Lyndon, P. E. Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ISBN: 3-540-41158-5.

A. Yu. Olshanskii, Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991. ISBN: 0-7923-1394-1.

R. Strebel, Appendix. Small cancellation groups. In: Sur les groupes hyperbolic d'après Mikhael Gromov (Bern, 1988), 227-273, Progr. Math. 83, Birkhäuser Boston, Boston, MA, 1990.
Voraussetzungen / BesonderesFamiliarity with very basic notions of group theory, definitions of free groups, group presentations, and graphs.