[Paper Review] The critical exponent, the Hausdorff dimension of the limit set and the convex core entropy of a Kleinian group

Kurt Falk, Katsuhiko Matsuzaki|arXiv (Cornell University)|Jun 22, 2011

Mathematical Dynamics and Fractals57 references8 citations

TL;DR

This paper establishes the relationship between three invariants of Kleinian groups: the critical exponent δ, the Hausdorff dimension dim_H of the limit set, and the convex core entropy h_c, showing δ ≤ dim_H ≤ h_c = upper box-counting dimension. It proves that h_c equals the upper box-counting dimension and provides conditions under which these inequalities are strict or equalities, particularly for geometrically tight groups and groups with bounded type.

ABSTRACT

In this paper we study the relationship between three numerical invariants associated to a Kleinian group, namely the critical exponent, the Hausdorff dimension of the limit set and the convex core entropy. The Hausdorff dimension of the limit set is naturally bounded below by the critical exponent and above by the convex core entropy. We investigate when these inequalities become strict and when they are equalities.

Motivation & Objective

• To clarify the relationship between the critical exponent δ, the Hausdorff dimension dim_H of the limit set, and the convex core entropy h_c of a Kleinian group.
• To establish that the convex core entropy h_c coincides with the upper box-counting dimension of the limit set, a new result in the literature.
• To investigate when the inequalities δ ≤ dim_H ≤ h_c are strict or become equalities, particularly in geometrically tight and bounded type groups.
• To characterize classes of Kleinian groups where δ < dim_H or dim_H < h_c, and to provide examples and conjectures based on partial results.
• To extend known results on geometrically finite and analytically finite groups to broader classes, including those satisfying tightness or bounded type conditions.

Proposed method

• Define the convex core entropy h_c as the critical exponent of an extended Poincaré series over a uniformly distributed set X within the convex hull H(Λ(G)) of the limit set.
• Prove that h_c equals the upper box-counting dimension (upper Minkowski dimension) of the limit set, establishing a new equality in the literature.
• Use the notion of ℓ-tightness, where every point in the convex hull is within distance ℓ of the boundary, to derive upper bounds s(n,ℓ) < n for h_c.
• Apply rough-isometry techniques and quasi-isometry arguments to show that certain hyperbolic surfaces constructed from graphs are not quasi-isometric to normal covers of convex compact surfaces.
• Leverage the existence of a uniform rough-isometry f on the universal cover that preserves a uniformly distributed set X and the group action, to prove that X is of bounded type.
• Use the structure of Cayley graphs and geometric group theory to show that if the convex core is quasi-isometric to a graph with infinite diameter and no finite-index subgroups, then the group is not of bounded type unless additional symmetry is present.

Experimental results

Research questions

• RQ1 When does the inequality δ < dim_H hold, and what geometric or dynamical conditions force this strict inequality?
• RQ2 Under what conditions does dim_H < h_c hold, and how does the geometry of the convex core influence this?
• RQ3 When is the convex core entropy h_c strictly less than n, and what role does ℓ-tightness play in this?
• RQ4 Can the bounded type condition be characterized geometrically, and how does it relate to the existence of symmetric structures in the convex core?
• RQ5 To what extent do the invariants δ, dim_H, and h_c coincide for finitely generated Kleinian groups in 3-dimensional hyperbolic space?

Key findings

• The convex core entropy h_c is proven to equal the upper box-counting dimension of the limit set, a novel result that unifies two previously distinct concepts.
• For any non-elementary Kleinian group, the inequality δ ≤ dim_H ≤ h_c holds, with the first inequality being classical and the second being a new general bound.
• For ℓ-tight Kleinian groups, there exists a constant s(n,ℓ) < n such that h_c < s(n,ℓ), showing that h_c < n even when the group is not of the first kind.
• The paper proves that h_c = dim_B holds for geometrically tight groups, extending known results on geometrically finite groups.
• A construction of a hyperbolic surface S from a k-regular graph shows that S is not quasi-isometric to any normal cover of a convex compact surface, implying that its fundamental group is not of bounded type unless symmetries force it.
• The existence of a uniform rough-isometry f on the universal cover preserving a uniformly distributed set X and the group action implies that X is of bounded type, leading to the conjecture that ℓ-tight groups of divergence type are of bounded type.