Entropy rigidity of negatively curved manifolds of finite volume

Peigné, Marc; Sambusetti, Andrea

doi:10.1007/s00209-018-2199-6

Cited by 5 publications

(4 citation statements)

References 32 publications

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“…[DPPS09]): we will come back to these examples at the end of the introduction. The topological entropy of the non-wandering set of the geodesic flow characterizes the hyperbolic metrics among Riemannian manifolds with pinched, negative curvature and with finite volume ( [PS19]).…”

Section: Lipschitz-topological Entropy Of the Geodesic Flowmentioning

confidence: 99%

Entropies of non positively curved metric spaces

Cavallucci¹

2021

Preprint

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We show the equivalences of several notions of entropy, such as a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform packing condition. Similar estimates will be given in case of closed subsets of the boundary of Gromovhyperbolic metric spaces with convex geodesic bicombings. A uniform Ahlfors regularity of the limit set of quasiconvex-cocompact actions on Gromov-hyperbolic packed metric spaces with convex geodesic bicombing will be shown, implying a uniform rate of convergence to the entropy. As a consequence we prove the continuity of the critical exponent for quasiconvex-cocompact groups with bounded codiameter.

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Section: Lipschitz-topological Entropy Of the Geodesic Flowmentioning

confidence: 99%

Entropies of non positively curved metric spaces

Cavallucci¹

2021

Preprint

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“…Finally, in order to prove Theorem 1.1, we need to recall a characterization of constant curvature spaces as those pinched, negatively curved spaces whose lattices realize the least possible value for the entropy. The minimal entropy problem has a long history and has been declined in many di↵erent ways so far; see [23], [4], [10] for the analogue of the following statement in the compact case, and [17] for a proof in the finite-volume case:…”

Section: Margulis Function For Regular Latticesmentioning

confidence: 99%

“…satisfying ( ) = (n 1)a. In the compact case, this result can be deduced from Knieper's work on spherical means (following the proof of Theorem 5.2, [24]), or from Bonk-Kleiner [4] (for convex-cocompact groups); on the other hand, see [17] for a complete proof in the case of non-uniform lattices and the analysis of the new di culties arising in the non-compact case.…”

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confidence: 97%

Asymptotic geometry of negatively curved manifolds of finite volume

Dal’Bo¹,

Peigné²,

Picaud³

et al. 2019

Ann. Sci. École Norm. Sup.

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We study the asymptotic behaviour of simply connected, Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice. If the quotient manifoldX = \X is asymptotically 1/4-pinched, we prove that is divergent and UX has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls B(x, R) in X is asymptotically equivalent to a purely exponential function c(x)e R , where is the topological entropy of the geodesic flow ofX. This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices in negatively curved spaces X (not asymptotically 1/4-pinched) where, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen-Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential.

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“…Some basic results about families of Riemannian metric and metric-measured spaces with uniformily bounded entropy, such as lemmas à-la-Margulis, finiteness and compactness results etc., are the object of [8]; see also [18] for an abelian version of the Margulis' lemma. Other local topological rigidity results under entropy bounds have recently appeared in [51] and [19].…”

Section: Introductionmentioning

confidence: 99%

Entropy and finiteness of groups with acylindrical splittings

Cerocchi

Sambusetti

2021

Groups Geom. Dyn.

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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 Ent.G; S/ < E, we have jSj 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 ı-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.

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Entropy rigidity of negatively curved manifolds of finite volume

Cited by 5 publications

References 32 publications

Entropies of non positively curved metric spaces

Entropies of non positively curved metric spaces

Asymptotic geometry of negatively curved manifolds of finite volume

Entropy and finiteness of groups with acylindrical splittings

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