The word problem for a finitely generated group G is the algorithmic problem of deciding whether a word in the generators represents the trivial element of G. When G is finitely presented, one can interpret this problem topologically by constructing a finite 2-complex X whose 1-cells and 2-cells correspond to the generators and relations. In this way, a word w in the generators which represents the trivial element of G will correspond to a loop in the 1-skeleton of the universal cover of X.
Geometry comes into the picture via the study of isoperimetric functions: An isoperimetric function is a function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation (or, equivalently, the area of a nullhomotopic loop in the complex described above in terms of the length of that loop). A Dehn function is an optimal isoperimetric function. Dehn functions can be understood as quantifying the complexity of the word problem.
This mini-course will survey what is known about Dehn functions for various classes of groups.
In the first lecture, we will discuss the precise connection between the solvability of the word problem for a group and the growth of its Dehn function. We will compute the Dehn functions of various classes of groups (introducing, on the way, hyperbolic groups and CAT(0) groups) and give examples of groups with very large Dehn functions.
In the second lecture, we will dive more deeply into the structure of Dehn functions, exploring the “isoperimetric spectrum”, i.e., the set {d | nd is the growth type of a Dehn function}, and computing some more examples.
We will wrap things up by discussing some generalisations and variations of Dehn functions, and how they connect to other algorithmic problems in geometric group theory. Time permitting, I will also tell you a bit about my own work in this direction.
These videos were part of the Geometric group theory without boundaries II virtual summer school.
