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 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.
Highlights
Take a geodesic triangle in the hyperbolic plane, and consider the rotations of angle π about the midpoints of the three sides, which we call the side involutions
Even if the area is not a rational multiple of π, torsion relations still appear for a dense choice of hyperbolic triangles
We are led to address the question whether or not relations that are not “consequences” of torsion relations can still be found for a dense set of triangles; we refer the reader to Definition 1.4 for a precise formulation
Summary
Take a geodesic triangle in the hyperbolic plane, and consider the rotations of angle π about the midpoints of the three sides, which we call the side involutions. Even if the area is not a rational multiple of π, torsion relations still appear for a dense choice of hyperbolic triangles (cf Theorem 5.2 below). Question A In the space of hyperbolic triangles whose area is fixed as φ ∈ (0, π), what are the necessary and sufficient conditions for the side involutions to generate the Coxeter group Z2 ∗ Z2 ∗ Z2? When ker ρξ contains a non-torsion-type word, the image ρξ (F2) cannot be isomorphic to a one-relator group with torsion with respect to the generating set {ρξ (X ), ρξ (Y )}. It is unclear to the authors whether the non-torsion-type kernel elements in F2 found in Theorem 1.5 can still be nontorsion-type in W Their additional property of being palindromic has a simple interpretation in W : they are products of two involutions in W (Lemma 4.1). The examples of pairs (θ, ξ ) exhibited therein have the property that the images of ρξ are isomorphic to fundamental groups of closed hyperbolic 3-manifolds
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