Andrea Sambusetti

Andrea Sambusetti
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Andrea Sambusetti

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Pub Categories

Mathematics - Differential Geometry (5)
Mathematics - Metric Geometry (3)
Mathematics - Dynamical Systems (1)
Mathematics - Geometric Topology (1)
Mathematics - Group Theory (1)

Publications Authored By Andrea Sambusetti

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. Read More

The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e. Read More

We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function. Read More

We consider the stable norm associated to a discrete, torsionless abelian group of isometries $\Gamma \cong \mathbb{Z}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\| \;\, \|_{st}$ and the distance $d$ is bounded by a constant only depending on the rank $n$ and on upper bounds for the diameter of $\bar X=\Gamma \backslash X$ and the asymptotic volume $\omega(\Gamma, d)$. We also prove that the upper bound on the asymptotic volume is equivalent to a lower bound for the stable systole of the action of $\Gamma$ on $(X,d)$; for this, we establish a Lemma \`a la Margulis for $\mathbb{Z}^n$-actions, which gives optimal estimates of $\omega(\Gamma,d)$ in terms of $\mathrm{stsys}(\Gamma,d)$, and vice versa, and characterize the cases of equality. Read More

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 $\tau(u)\ex^{Er}$ for the volume-density of geodesic spheres (with $\tau$ constant in case $DR_M$ is bounded). Read More

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 . Read More