Geometric group theory

The graphic work of M. C. Escher is much appreciated by mathematicians and is also enjoyed by a much wider public. A quick look at Escher’s pictures of tesselations of the Euclidean and hyperbolic planes makes everyone see that these two geometries have different asymptotic behavior: the Euclidean plane has quadratic polynomial growth, while the hyperbolic plane has exponential growth. The reviewer has found, in teaching courses to audiences not at all inclined to mathematics, that the difference between these two geometries can be immediately understood and appreciated by looking at these pictures. The evident quadratic and exponential growth that can be seen in these pictures is a special case of one of the earliest and most easily understandable asymptotic invariant of infinite (and finitely generated) groups, namely, the growth of a group. Together with the concept of growth came the idea of considering a group to be a metric space and to relate this metric space to more conventional metric spaces on which the group acts by the concept of quasi-isometry. Namely, if Γ is a finitely generated group and x ∈ Γ, define |x|, the distance of x from the identity, relative to a set of generators x1, ··· , xn for Γ, to be the minimum length of a word in x±1 1 , ··· , x±1 n expressing the element x ∈ Γ. The distance between x, y ∈ Γ is defined to be |x−1y|. It is easily checked that this distance makes Γ into a metric space and that different (finite) sets of generators give metrics which are bi-Lipschitz equivalent. The growth function of Γ is the function on the natural numbers that to n associates the cardinality of the ball of radius n, i.e., the number of elements in Γ expressible as words of length at most n. The growth of Γ is the order of growth of this function