Classical Schottky characterization for low-critical-exponent Kleinian groups

Determine whether every finitely generated Kleinian group with critical exponent less than one is of classical Schottky type, extending the established virtual-freeness and geometric-finiteness conclusions.

Background

The paper proves that finitely generated discrete groups acting on proper CAT(-1) spaces with critical exponent less than one are geometrically finite and free under a torsion-freeness assumption, and consequently that finitely generated Kleinian groups with critical exponent less than one are geometrically finite and virtually free. The authors note that Kapovich posed a stronger question for Kleinian groups: whether these groups must have classical Schottky structure rather than merely virtual freeness and geometric finiteness.

The problem remains outside the results established in the paper, which explicitly states that it does not pursue this question and cites related work for partial developments.

References

In Problem 1.6 and Conjecture 10.14, Kapovich asked further whether every finitely generated Kleinian group with critical exponent less than one must be of classical Schottky type. This is stronger than the virtual-freeness and geometric-finiteness conclusions proved above. We do not pursue the question here, and we refer to for related results.

— Actions on CAT(-1) spaces with critical exponent less than 1  (2608.18906 - Liu et al., 19 Aug 2026) in Remark following Corollary 1.3 in Section 1, Introduction

We do not know if the condition $\delta(\Gamma)<1$ is sharp. There are geometrically infinite surface Kleinian groups constructed by whose critical exponent is $2$, so the sharp constant should lie in $[1,2]$. Motivated by this, we ask the following question. What is the smallest real number $\delta_0\geq 1$ such that the following statement holds? If $\Gamma$ is a finitely generated, torsion-free, discrete group acting on a proper $CAT(-1)$ space whose critical exponent is $<\delta_0$, then $\Gamma$ must be geometrically finite.

— Actions on CAT(-1) spaces with critical exponent less than 1  (2608.18906 - Liu et al., 19 Aug 2026) in Subsection “The structure of limit sets,” Section 1, Introduction; Question environment immediately following the discussion