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Andrea Sambusetti

Publications and source records attributed to Andrea Sambusetti.

16 recordsLinked to original sources
arXiv2026

CAT(0) spaces quasi-isometric to Euclidean spaces

We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to R^n, which are not homeomorphic to it. We prove that our example is sharp in a suitable sense. Finally, we provide an example of a sequence of proper, geodesically complete, CAT(0) spaces that are not homology manifolds and that converge in the Gromov-Hausdorff sense to a topological manifold: this shows that the set of topological manifolds is not open in the class of proper, geodesically complete, CAT(0) spaces.

math.MG↗
arXiv2024

Finiteness of CAT(0) group actions

We prove some finiteness results for discrete isometry groups $Γ$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = Γ\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $Γ$ is bounded above by a universal constant only depending on the packing constants and the codiameter. The main tool is a splitting theorem for sufficiently collapsed actions: namely we show that if a geodesically complete, packed, CAT$(0)$-space admits a discrete, cocompact group of isometries with sufficiently small systole then it necessarily splits a non-trivial Euclidean factor.

math.GR↗
arXiv2024

Convergence and collapsing of CAT$(0)$-lattices

We study the theory of convergence for CAT$(0)$-lattices (that is groups $Γ$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces) and their quotients (CAT$(0)$-orbispaces). We describe some splitting and collapsing phenomena, explaining precisely how these action can degenerate to a possibly non-discrete limit action. Finally, we prove a compactness theorem for the class of compact CAT$(0)$-homology orbifolds, and some applications: an isolation result for flat orbispaces and an entropy-pinching theorem.

math.MG↗
arXiv2024

Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

We prove a quantitative version of the classical Tits' alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.

math.MG↗
arXiv2022

Thin actions on CAT(0) spaces

We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.

math.MG↗
arXiv2021

Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free $δ$-hyperbolic marked groups whose entropy is bounded above by a number $H$ is finite with cardinality depending on $δ$ and $H$ alone. From these results, we draw homotopical and topological finiteness theorems for compact metric spaces and manifolds.

math.GR↗
arXiv2021

Packing and doubling in metric spaces with curvature bounded above

We study locally compact, locally geodesically complete, locally CAT(k) spaces (GCBA(k)-spaces). We prove a Croke-type local volume estimate only depending on the dimension of these spaces. We show that a local doubling condition, with respect to the natural measure, implies pure-dimensionality. Then, we consider GCBA(k)-spaces satisfying a uniform packing condition at some fixed scale or a doubling condition at arbitrarily small scale, and prove several compactness results with respect to pointed Gromov-Hausdorff convergence. Finally, as a particular case, we study convergence and stability of M^k-complexes with bounded geometry.

math.MG↗
arXiv2020

Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces

We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is hyperbolic in the sense of Gromov and the lower bound of the curvature by an upper bound on the entropy which we recall the definition.

math.DG↗
arXiv2018

Growth gap in hyperbolic groups and amenability

We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.

math.GR↗
arXiv2018

Entropy and finiteness of groups with acylindrical splittings

We prove that there exists a positive, explicit function $F(k, E)$ such that, for any group $G$ admitting a $k$-acylindrical splitting and any generating set $S$ of $G$ with $\mathrm{Ent}(G,S)<E$, we have $|S| \leq F(k, E)$. We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, $D$-quasiconvex $k$-malnormal amalgamated products acting on $δ$-hyperbolic spaces or on $CAT(0)$-spaces with entropy bounded by $E$. A number of finiteness results for interesting families of Riemannian or metric spaces with bounded entropy and diameter also follow: Riemannian 2-orbifolds, non-geometric $3$-manifolds, higher dimensional graph manifolds and cusp-decomposable manifolds, ramified coverings and, more generally, CAT(0)-groups with negatively curved splittings.

math.MG↗
arXiv2017

Local topological rigidity of non-geometric $3$-manifolds

We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. We prove a lower bound "à la Margulis" for the systole and a volume estimate for these manifolds, only in terms of an upper bound of entropy and diameter. We then deduce corresponding local topological rigidy results in the class $\mathscr{M}_{ngt}^\partial (E,D) $ of compact non-geometric 3-manifolds with torsionless fundamental group (with possibly empty, non-spherical boundary) whose entropy and diameter are bounded respectively by $E, D$. For instance, this class locally contains only finitely many topological types; and closed, irreducible manifolds in this class which are close enough (with respect to $E,D$) are diffeomorphic. Several examples and counter-examples are produced to stress the differences with the geometric case.

math.MG↗
arXiv2015

Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary

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., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian distance $d$ on $\mathbb H$ there is a unique subRiemanniann metric $d'$ for which $d-d'$ goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group.

math.DG↗
arXiv2015

Quantitative Bounded Distance Theorem and Margulis' Lemma for Z^n actions with applications to homology

We consider the stable norm associated to a discrete, torsionless abelian group of isometries $Γ\cong \mathbb{Z}^n$ 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 $\bar X=Γ\backslash X$ and the asymptotic volume $ω(Γ, d)$. 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 $Γ$ on $(X,d)$; for this, we establish a Lemma à la Margulis for $\mathbb{Z}^n$-actions, which gives optimal estimates of $ω(Γ,d)$ in terms of $\mathrm{stsys}(Γ,d)$, and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters $n, \mathrm{diam}(\bar X)$ and $ω(Γ, d)$ (or $\mathrm{stsys} (Γ,d)$) are necessary to bound the difference $d -\| \;\, \|_{st}$, by providing explicit counterexamples for each case. As an application, we prove that the number of connected components of any optimal, integral $1$-cycle in a closed Riemannian manifold $\bar X$ either is bounded by an explicit function of the first Betti number, $ \mathrm{diam}(\bar X)$ and $ω(H_1(\bar X, \mathbb{Z}), d)$, or is a sublinear function of the mass.

math.MG↗
arXiv2009

Submanifolds, Isoperimetric Inequalities and Optimal Transportation

The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the one obtained by J. Michael and L. Simon. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension.

math.DG↗
arXiv2009

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 .

math.DS↗