Shapes of hyperbolic triangles and once-punctured torus groups

TL;DR

This paper investigates the geometric and algebraic properties of hyperbolic triangles and once-punctured torus groups, revealing generic conditions for trivial relations and dense sets with nontrivial relations, and studying deformation spaces of hyperbolic structures.

Contribution

It establishes generic conditions for the absence of nontrivial relations in certain hyperbolic triangle groups and analyzes deformation spaces of hyperbolic torus structures with cone points.

Findings

Most hyperbolic triangles have no nontrivial relations in their associated groups.

A dense set of triangles with fixed area admits nontrivial relations mapping to hyperbolic translations.

Concrete examples show hyperbolic 3-manifold groups can arise as images of holonomy representations.

Abstract

Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $\varphi\in(0,\pi)\setminus\mathbb{Q}\pi$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_\theta$ of singular hyperbolic metrics on a torus with a single cone point of angle $\theta=2(\pi-\varphi)$, and answer an analogous question for the holonomy map $\rho_\xi$ of such a hyperbolic structure $\xi$. In an appendix by X.~Gao, concrete examples of $\theta$ and $\xi\in\mathfrak{C}_\theta$ are given where the image of each $\rho_\xi$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.