Growth tightness in group theory and Riemannian geometry

This paper is a survey on the "growth tightness" results for discrete groups and fundamental groups of negatively curved manifolds, based on original work of the author. We are interested in the exponential growth rate (also known as the entropy Ent(G)) of discrete group G, endowed with a left invariant metric; the metric may as well be a word metric or, when G acts freely by isometries on a Riemannian manifold X, the Riemannian distance between points of an orbit Gx (i.e. the Riemannian lenght of geodesic loops at x). We begin giving an elementary proof of growth tightness of free groups G, implying a well-known asymptotic characterization of free groups: for any group G on set A of k generators, we have Ent(G) < log(2k-1), with equality if and only if G is free on A. We also give in this case a precise estimate of the entropy gap Ent(G)-Ent(G/N) between the entropy of G and the entropy of any quotient G/N, in terms of the length of the relations. In the second part, we give similar statements for free and amalgamated products, introducing and estimating the entropy of the space Lc(G) of "weighted words" on G, to circumvent the difficulty of working directly with minimal reduced words in the general case. Section 3 contains the analogue of these results for fundamental groups G of a compact closed hyperbolic surface S, translating the growth tightness condition in terms of Galois coverings S' of the surface S; we also give estimates for the entropy gap Ent(H^2)-Ent(S') in terms of the systole of the covering S'. In section 4, we generalize these results to general, closed, negatively curved Riemannian manifolds.