Geometric finiteness threshold below critical exponent two

Establish whether, for some or all proper CAT(-1) spaces, every finitely generated torsion-free discrete group with critical exponent less than two is geometrically finite, as conjectured by Kapovich.

Background

The paper establishes geometric finiteness under the stricter hypothesis that the critical exponent is less than one. It discusses Kapovich's conjecture that the threshold could be increased to two, noting that the claim is known in dimension three for Kleinian groups but is unresolved in the general range from one to two.

The authors explicitly state that their method fails in this broader range, so the conjectured geometric-finiteness implication is not settled by the paper.

References

Kapovich also conjectured Conjecture 10.17 that under a milder condition $\delta(\Gamma)<2$, it is sufficient to imply the geometric finiteness of $\Gamma$. This is true for $n=3$, which follows from the tameness Theorem, the Ahlfors measure conjecture, and the work of Bishop–Jones (See also ). However, the current method fails in the general range of $1\leq\delta(\Gamma)<2$.

— 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