[Paper Review] On the minimum dilatation of pseudo-Anosov homeomorphisms on surfaces of small genus
This paper determines the minimum dilatation of pseudo-Anosov homeomorphisms on orientable surfaces of genus 3, 4, and 5, proving it equals the largest real root of specific degree-2g polynomials that are minimal Salem numbers. It provides a lower bound for genus 6–8 and offers a simplified, elementary proof of Cho and Ham's result for genus 2, establishing uniqueness up to conjugacy and symmetries for the minimal dilatation case.
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus g=2 to 5, the mimimum dilatation is the smallest Salem number for polynomials of degree 2g.
Motivation & Objective
- To compute the minimum dilatation δ⁺_g for pseudo-Anosov homeomorphisms on orientable surfaces of genus g = 3, 4, 5.
- To establish a lower bound for δ⁺_g when g = 6, 7, 8.
- To provide a simplified, elementary proof of the minimal dilatation for genus 2, previously established by Cho and Ham.
- To confirm that the minimal dilatation for g = 2 to 5 corresponds to the smallest known Salem number for degree-2g polynomials.
- To investigate the uniqueness of the conjugacy class realizing the minimal dilatation in the mapping class group Mod(g), up to hyperelliptic involution and covering transformations.
Proposed method
- The authors use Rauzy–Veech induction on translation surfaces to construct explicit examples of pseudo-Anosov maps with minimal dilatation.
- They apply Dehn twists on a surface with a specific permutation and path to generate the mapping class and compute its action on homology.
- The dilatation is derived from the Perron root of the characteristic polynomial of the monodromy matrix associated with the mapping class.
- The construction relies on partitioning the surface into polygonal pieces and tracking translation vectors between the original and transformed surface.
- For genus 2, the method uses two sequences of Dehn twists whose action on homology yields the same characteristic polynomial X⁴ - X³ - X² - X + 1.
- The proof leverages properties of Salem numbers and the fact that the minimal dilatation corresponds to the smallest Mahler measure among degree-2g integral polynomials.
Experimental results
Research questions
- RQ1What is the minimum dilatation of a pseudo-Anosov homeomorphism on a genus 3 surface with an orientable invariant foliation?
- RQ2How does the minimal dilatation for genus 4 and 5 compare to known Salem numbers and polynomials of degree 2g?
- RQ3Can the minimal dilatation for genus 2 be reproven with a simpler, more elementary method than prior train track-based approaches?
- RQ4Is the conjugacy class realizing the minimal dilatation unique in Mod(g) for g = 2 to 5, up to symmetries like the hyperelliptic involution?
- RQ5What is the smallest possible dilatation for genus 6 to 8, and can a lower bound be established?
Key findings
- For genus 3, the minimum dilatation δ⁺₃ is the largest real root of X⁶ - X⁴ - X³ - X² + 1, approximately 1.40127.
- For genus 4, the minimum dilatation δ⁺₄ is the largest real root of X⁸ - X⁵ - X⁴ - X³ + 1, approximately 1.28064.
- For genus 5, the minimum dilatation δ⁺₅ is the largest real root of X¹⁰ + X⁹ - X⁷ - X⁶ - X⁵ - X⁴ - X³ + X + 1, approximately 1.17628.
- All minimal dilatations for g = 2 to 5 are Salem numbers, and their defining polynomials have the smallest known Mahler measure for degree 2g.
- The minimal dilatation for genus 2 is confirmed to be the largest root of X⁴ - X³ - X² - X + 1, with δ₂ ≈ 1.72208, and the proof is simplified using Dehn twist sequences.
- For genus 6 to 8, the paper establishes a lower bound for the minimal dilatation, though the exact value remains open.
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This review was created by AI and reviewed by human editors.