Andrea Sambusetti

Growth tightness of free and amalgamated products

Abstract We show that every nontrivial free product, different from the infinite dihedral group, is growth tight with respect to any algebraic distance: that is, its exponential growth rate is strictly greater than the corresponding growth rate of any of its proper quotients. A similar property holds for the amalgamated product of residually finite groups over a finite subgroup. As a consequence, we provide examples of finitely generated groups of uniform exponential growth whose minimal growth is not realized by any generating set.

Growth tightness of surface groups

Abstract We prove that fundamental groups of closed oriented surfaces ∑g of genus g ≥ 2 are growth tight with respect to hyperbolic metrics and to the word metric relative to their canonical presentation: this means that the exponential growth rate of π1 (∑g, with respect to these metrics, is always strictly greater than the corresponding growth rate of any of its proper quotients. As an application, we give a new, purely analytic proof of Hopficity of surface groups.

On the growth of nonuniform lattices in pinched negatively curved manifolds

We study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold .

On Minimal Entropy and Stability

We study n-manifolds Y whose fundamental groups are subexponential extensions of the fundamental group of some closed locally symmetric manifold X of negative curvature. We show that, in this case, MinEnt(Y)n is an integral multiple of MinEnt(X)n, and the value MinEnt(Y) is generally not attained (unless if Y is diffeomorphic to X). This gives a new class of manifolds for which the minimal entropy problem is completely solved. Several examples (even complex projective), obtained by gluings and by taking plane intersections in complex projective space, are described. Some problems about topological stability, related to the minimal entropy problem, are also discussed.

On asymptotically harmonic manifolds of negative curvature

We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan–Hadamard spaces of strictly negative curvature, by the existence of an asymptotic equivalent

Convergence and Counting in Infinite Measure@@@Convergence et comptage en mesure infinie

\tau (u)\mathrm {e}^{Er}

On the horoboundary and the geometry of rays of negatively curved manifolds

τ(u)eEr for the volume-density of geodesic spheres (with