On the growth of quotients of Kleinian groups

Abstract

Abstract. We study the growth and divergence of quotients of Kleinian groups G (i.e. discrete, torsionless groups of isometries of a Cartan-Hadamard manifold with pinched negative curvature). Namely, we give general criteria ensuring the divergence of a quotient groupḠ of G and the 'critical gap property' δḠ < δ G . As a corollary, we prove that every geometrically finite Kleinian group satisfying the parabolic gap condition (i.e. δ P < δ G for every parabolic subgroup P of G) is growth tight. These quotient groups naturally act on non-simply connected quotients of a Cartan-Hadamard manifold, so the classical arguments of Patterson-Sullivan theory are not available here; this forces us to adopt a more elementary approach, yielding as by-product a new elementary proof of the classical results of divergence for geometrically finite groups in the simply connected case. We construct some examples of quotients of Kleinian groups and discuss the optimality of our results.

Cogrowth for group actions with strongly contracting elements

“…By definition, o ∈ Z 0 and z ∈ Z 2m . By (11), π Z 2m (o) is close to z 2m . By Corollary 2.1, any geodesic from o to z ends with a segment that stays close to the subsegment of Z 2m between z 2m and z = z 2m .…”

“…For relatively hyperbolic groups the complementary growth gap specializes to the parabolic growth gap of [11], which requires that the growth of parabolic subgroups of a relatively hyperbolic group is strictly less than the growth rate of the whole group. For another non-cocompact example, we showed in [2] that the action of the mapping class group of a hyperbolic surface on its Teichmüller space has complementary growth gap.…”

“…When p is sufficiently large, d(z i , z i ) 10K for all i, so (11) implies that Z i Z j for all 0 i < j 2k, where is the order of Proposition 2.3 on Y[Z 0 , Z 2k ].…”

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

“…Since their introduction, the property of growth-tightness was then established by Arzhantseva and Lysenok [3] for hyperbolic groups, by Sambusetti [71] [72] for free products and cocompact Kleinian groups, and by Dal'bo, Peigné, Picaud and Sambusetti [29] for geometrically finite Kleinian groups with parabolic gap property. The present author [78] realized their most arguments in a broad setting and showed growth-tightness of groups with non-trivial Floyd boundary, subsequently generalizing the result to any group acting properly and cocompactly on a geodesic metric space with a contracting element.…”

“…To present the criterion, one needs to introduce the Poincaré series d G (1, g)), s ≥ 0, which clearly diverges for s < ω d G (Γ) and converges for s > ω d G (Γ). Dalbo et al [29] presented a very useful criterion to differentiate the critical exponents of two Poincaré series, which is the key tool to establish growth-tightness. We formulate it with purpose to exploit the extension map.…”