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September 23, 2025Open Access

Peripheral birationality for 3-dimensional convex co-compact PSL 2 ℂ varieties

Key Points

The peripheral map is shown to be a birational isomorphism, indicating a one-to-one relationship in character varieties.
Key findings demonstrate the image of the map is generically one-to-one, informed by earlier research on hyperbolic 3-manifolds.
The analysis incorporates the Bonahon–Schläfli formula, shedding light on volume rigidity in discrete co-compact representations.
This work generalizes previous results in the field, bridging findings in one-cusped and finite volume hyperbolic 3-manifolds.

Abstract

Let M be a hyperbolizable 3-manifold with boundary, and let χ 0 (M) be a component of the PSL 2 ℂ-character variety of M that contains the convex co-compact characters. We show that the peripheral map i * :χ 0 (M)→χ(∂M) to the character variety of ∂M is a birational isomorphism with its image, and in particular is generically a one-to-one map. This generalizes work of Dunfield (one cusped hyperbolic 3-manifolds) and Klaff–Tillmann (finite volume hyperbolic 3-manifolds). We use the Bonahon–Schläfli formula and volume rigidity of discrete co-compact representations.

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