Questions in Geometric Group Theory — Q 5.7

Definitions and mathematical statement

For an integer $n\ge2$, let $\mathbb H^n$ be real hyperbolic space with ideal boundary $\partial\mathbb H^n\cong S^{n-1}$. For a finitely generated discrete subgroup $\Gamma\le\operatorname{SO}(n,1)$, its limit set $\Lambda(\Gamma)$ is the set of accumulation points in $\partial\mathbb H^n$ of an orbit $\Gamma o$, $o\in\mathbb H^n$. Kapovich asks separately whether there exist such $n,\Gamma$ for which the boundary action $\Gamma\curvearrowright\Lambda(\Gamma)$ is (i) not ergodic, or (ii) not recurrent.

Both properties require a measure class, absent from the literal question. For example, if $\mu$ is the restriction of spherical Lebesgue measure to $\Lambda(\Gamma)$, ergodicity means that every measurable set $A\subseteq\Lambda(\Gamma)$ invariant modulo null sets satisfies $\mu(A)=0$ or $\mu(\Lambda(\Gamma)\setminus A)=0$. A usual recurrence convention for a nonsingular countable-group action requires that for every measurable $A$ of positive measure, almost every $\xi\in A$ has infinitely many distinct $\gamma\in\Gamma$ with $\gamma\xi\in A$. These give a precise conditional reading of the two existence questions; they do not resolve which measure and recurrence convention the source intends. When $\mu(\Lambda(\Gamma))=0$, these restricted-measure conditions are vacuous, so a nontrivial example under this reading must have positive-measure limit set.

The supplied research summary reports negative answers to both existence questions in dimensions $n\le3$: recurrence attributed to Ahlfors (1980), and ergodicity for positive-measure limit sets attributed to the Ahlfors measure conjecture through Canary (1993) and Agol (2004)/Calegari--Gabai (2004). It reports both questions open for $n\ge4$ and related to the higher-dimensional Ahlfors measure conjecture. Its analogy with nonergodic recurrent $\operatorname{PU}(2,1)$ actions having $\Lambda=S^3$ is not an example in $\operatorname{SO}(n,1)$. It attributes a conjecture of higher-dimensional nonergodic examples to the question's author. These are supplied triage claims, whose precise hypotheses and measure conventions need review.

Short English statement

Can a finitely generated discrete real-hyperbolic group act nonergodically, or nonrecurrently, on its limit set? The measure and meaning of recurrence must be specified. The supplied triage distinguishes low-dimensional negative results from higher-dimensional open questions.

Sources

• Local UnsolvedMath registry, permanent ID $1100507$, exact source code catalogue entry, "Questions in Geometric Group Theory --- Q 5.7."
• Bestvina, Questions in Geometric Group Theory (2004), Question 5.7, PDF page 14; attributed to Misha Kapovich. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf; manual transcription; recorded access 2026-07-29.
• Supplied research summary names Ahlfors (1980), Canary (1993), Agol (2004), and Calegari--Gabai (2004), and describes the Ahlfors measure conjecture and complex-hyperbolic examples. Complete bibliographic citations for these claims and examples are not supplied.