Contact Entropy and Fundamental-Group Growth

Definitions and mathematical statement

The supplied question concerns a contact three-manifold $(Y,\xi)$. Its intended scope requires clarification: the record does not specify whether $Y$ is closed or connected, whether $\xi$ is coorientable, or which entropy convention applies on a noncompact manifold.

Under the explicitly conditional interpretation that $Y$ is a smooth, closed, connected three-manifold and $\xi$ is a smooth cooriented contact structure, a contact form defining $\xi$ is a smooth one-form $\alpha$ with $\ker\alpha=\xi$, compatible with the chosen coorientation, and satisfying $\alpha\wedge{\,\mathrm{d}}\alpha\ne0$ everywhere. Its Reeb vector field $R_\alpha$ is uniquely determined by \[ \alpha(R_\alpha)=1,\qquad {\,\mathrm{d}}\alpha(R_\alpha,\cdot)=0. \] Write $\varphi_\alpha^t$ for its flow and define its topological entropy to be $h_{\mathrm{top}}(\varphi_\alpha^1)$. Closedness ensures that the flow is complete and that the time-one map acts on a compact space. The positive-entropy condition in the source is \[ \forall\alpha\text{ defining }\xi,\qquad h_{\mathrm{top}}(\varphi_\alpha^1)>0. \] This condition does not assert a positive lower bound uniform over all such forms.

For a finitely generated group $G$ and a finite symmetric generating set $S$, let $\lvert g\rvert_S$ be word length and set \[ B_S(n)=\{g\in G:\lvert g\rvert_S\le n\}. \] The group has exponential growth if \[ \liminf_{n\to\infty}\frac{\log\lvert B_S(n)\rvert}{n}>0; \] this property is independent of the finite generating set. Under the closed, connected interpretation, $\pi_1(Y,y_0)$ is finitely generated, and its growth type is independent of the basepoint $y_0$.

The question, under these stated conventions, is whether every such $(Y,\xi)$ satisfies \[ \left[\forall\alpha\text{ defining }\xi,\quad h_{\mathrm{top}}(\varphi_\alpha^1)>0\right] \quad\Longrightarrow\quad \pi_1(Y,y_0)\text{ has exponential growth}. \] The closedness, connectedness, smoothness, and coorientation conventions above make this formulation precise but are not explicit hypotheses in the supplied record. Their agreement with the intended source formulation requires human review; the scope beyond these conventions remains unresolved.

Short English statement

The question asks whether positive topological entropy for every Reeb flow of a contact three-manifold forces its fundamental group to have exponential growth. The precise formulation given uses explicitly conditional closed, connected, smooth, cooriented conventions; the supplied record does not settle these hypotheses. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied permanent UnsolvedMath ID $30003529$, title "Contact Entropy and Fundamental-Group Growth," exact source problem code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Dynamische Systeme (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/32, pp. 1987--2046. https://doi.org/10.4171/owr/2017/32. The background describes the reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Alves, Cylindrical contact homology and entropy. https://arxiv.org/abs/1410.3380.
• Supplied literature source: Oberwolfach Report 32/2017. https://doi.org/10.4171/owr/2017/32.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that known positive-entropy contact three-manifolds have exponentially growing fundamental group, while the general implication remains a conjecture without a general proof or counterexample. This is supplied dated curation derived from the OWR material, not independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.