# Van Kampen Diagram - Generalizations and Other Applications

Generalizations and Other Applications

• There are several generalizations of van-Kampen diagrams where instead of being planar, connected and simply connected (which means being homotopically equivalent to a disk) the diagram is drawn on or homotopically equivalent to some other surface. It turns out, that there is a close connection between the geometry of the surface and certain group theoretical notions. A particularly important one of these is the notion of an annular van Kampen diagram, which is homotopically equivalent to an annulus. Annular diagrams, also known as conjugacy diagrams, can be used to represent conjugacy in groups given by group presentations. Also spherical van Kampen diagrams are related to several versions of group-theoretic asphericity and to Whitehead's asphericity conjecture, Van Kampen diagrams on the torus are related to commuting elements, diagrams on the real projective plane are related to involutions in the group and diagrams on Klein's bottle are related to elements that are conjugated to their own inverse.
• Van Kampen diagrams are central objects in the small cancellation theory developed by Greendlinger, Lyndon and Schupp in the 1960s-1970s. Small cancellation theory deals with group presentations where the defining relations have "small overlaps" with each other. This condition is reflected in the geometry of reduced van Kampen diagrams over small cancellation presentations, forcing certain kinds of non-positively curved or negatively cn curved behavior. This behavior yields useful information about algebraic and algorithmic properties of small cancellation groups, in particular regarding the word and the conjugacy problems. Small cancellation theory was one of the key precursors of geometric group theory, that emerged as a distinct mathematical area in lated 1980s and it remains an important part of geometric group theory.
• Van Kampen diagrams play a key role in the theory of word-hyperbolic groups introduced by Gromov in 1987. In particular, it turns out that a finitely presented group is word-hyperbolic if and only if it satisfies a linear isoperimetric inequality. Moreover, there is an isoperimetric gap in the possible spectrum of isomperimetric functions for finitely presented groups: for any finitely presented group either it is hyperbolic and satisfies a linear isoperimetric inequality or else the Dehn function is at least quadratic.
• The study of isoperimetric functions for finitely presented groups has become an important general theme in geometric group theory where substantial progress has occurred. Much work has gone into constructing groups with "fractional" Dehn functions (that is, with Dehn functions being polynomials of non-integer degree). The work of Rips, Ol'shanskii, Birget and Sapir explored the connections between Dehn functions and time complexity functions of Turing machines and showed that an arbitrary "reasonable" time function can be realized (up to appropriate equivalence) as the Dehn function of some finitely presented group.
• Various stratified and relativized versions of van Kampen diagrams have been explored in the subject as well. In particular, a stratified version of small cancellation theory, developed by Ol'shanskii, resulted in constructions of various group-theoretic "monsters", such as the Tarski Monster, and in geometric solutions of the Burnside problem for periodic groups of large exponent. Relative versions of van Kampen diagrams (with respect to a collection of subgroups) were used by Osin to develop an isoperimetric function approach to the theory of relatively hyperbolic groups.