Research Problems in Function Theory — Problem 5.52

Definitions and mathematical statement

Let $\Gamma$ be a Fuchsian group, meaning a discrete subgroup of the holomorphic automorphism group of $\mathbb D$. It acts also on $\mathbb T=\partial\mathbb D$. Suppose a Borel set $B\subset\mathbb T$ has pairwise disjoint translates in the sense \[ B\cap\gamma(B)=\varnothing\qquad (\gamma\in\Gamma\setminus\{\mathrm{id}\}). \] For compact $K\subset\mathbb C$, its logarithmic capacity is $\operatorname{cap}K=\exp[-\inf_\mu\iint\log(1/|z-w|)\,d\mu(z)d\mu(w)]$, where $\mu$ ranges over probability measures supported on $K$. Positive capacity of $B$ means some compact subset has positive capacity. Is $\operatorname{cap}B>0$ sufficient to force $\Gamma$ to be of convergence type, namely \[ \sum_{\gamma\in\Gamma}\left(1-|\gamma(0)|\right)<\infty? \] Equivalently one may use $\sum_\gamma e^{-d_{\mathbb D}(0,\gamma(0))}$ with the usual curvature-minus-one hyperbolic distance. Finite stabilizers do not affect convergence. The source gives the converse: convergence-type groups admit such a positive-capacity set. The requested implication therefore asks whether the existence of a sufficiently large disjoint-translates boundary set characterizes convergence type.

Short English statement

Does a boundary set of positive logarithmic capacity with disjoint Fuchsian-group translates force convergence of the group's Poincaré series?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305052 (AMR-022-5052), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.52. Original source indexed by AMR-022-5052. https://arxiv.org/abs/1809.07200.