On the growth of nonuniform lattices in pinched negatively curved manifolds

Abstract

International audienceWe 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

Le flot géodésique des quotients géométriquement finis des géométries de Hilbert

“…Si le groupe est "trop petit", cette égalité devient en général fausse, et on a seulement δ h vol . Dans [Dal'Bo et al 2009], Françoise Dal'bo, Marc Peigné, Jean-Claude Picaud et Andrea Sambusetti ont étudié cette question pour les sous-groupes de covolume fini de variétés de Hadamard, à courbure négative pincée. Ils ont montré le résultat suivant.…”

On the growth of quotients of Kleinian groups

“…Recall that the volume entropy of a covering X and the critical exponent of its automorphism groupḠ are the asymptotic invariants defined respectively as ω(X ) = lim sup where B X (x, R) denotes the ball of radius R in X centred atx; see [2,7,11,15]. Both numbers are independent of the choice of the pointx ∈X ; moreover, the critical exponent δḠ coincides with the abscissa of convergence of the Poincaré series ofḠ, PḠ(s,x) = ḡ∈Ḡ e −sd(x,ḡx) , s ∈ R + , and, if k X ≥ −K 2 0 , we know that δḠ ≤ ω(X ) ≤ (n − 1)K 0 < +∞ by the standard comparison theorems of Riemannian geometry.…”

“…Both numbers are independent of the choice of the pointx ∈X ; moreover, the critical exponent δḠ coincides with the abscissa of convergence of the Poincaré series ofḠ, PḠ(s,x) = ḡ∈Ḡ e −sd(x,ḡx) , s ∈ R + , and, if k X ≥ −K 2 0 , we know that δḠ ≤ ω(X ) ≤ (n − 1)K 0 < +∞ by the standard comparison theorems of Riemannian geometry. The invariants ω(X ) and δḠ (hence, the aforementioned spectra Ent(X 0 ) and Exp(X 0 )) coincide if X 0 is compact, but they may differ when X 0 has finite volume (see [7]). The critical exponent is considerably more interesting than volume entropy when G has infinite volume: for instance, for every geometrically finite hyperbolic manifold X 0 = G\H of infinite volume, the spectrum of entropies Ent(X 0 ) is always reduced to the value n − 1 (since X 0 and all its coverings contain a funnel); on the other hand, the spectrum of critical exponents Exp(X 0 ) is a particular subset of the interval [0, n − 1] which strongly depends on X 0 , and whose top value δ G is related to the dynamics of the geodesic flow on UX 0 .…”

Entropy Rigidity for Metric Spaces

“…(See [28], [13], [33] for proofs.) More recently, F. Dalbo, M. Peigne, J. Picaud, and A. Sambusetti showed that if M is non-compact but 1/4-pinched (i.e.…”

“…when −b 2 ≤ κ(M ) ≤ −a 2 and b 2 /a 2 ≤ 4), then the above theorem still holds. They also showed that 1/4 is optimal by constructing examples of non-compact (1/4 + ε)-pinched manifold for which the volume entropy is strictly larger than the critical exponent [13].…”