On the spectrum of the number of geodesics and tight geodesics in the curve complex

2025/05/03 by Ryo Matsuda, Matsuda, Ryo, Kanako Oie +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57K20 #Secondary 57K99 #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2505.01801

openalex publication_date 2025/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be an oriented surface of type (g, n). We are interested in geodesics in the curve complex \mathcal C(S) of S. In general, two 0-simplexes in \mathcal C(S) have infinitely many geodesics connecting the two simplexes while another geodesics called tight geodesics are always finitely many. On the other hand, we may find two 0-simplexes in \mathcal C(S) so that they have only finitely many geodesics between them. In this paper, we consider the spectrum of the number of geodesics with length d (≥ 2) in \mathcal C(S) and tight geodesics, which is denoted by \mathfrakSp_d(S) and \mathfrakSp_d^T(S), respectively. In our main theorem, it is shown that \mathfrakSp_d(S) ⊂ \mathfrakSp_d^T(S) in general, but \mathfrakSp₂(S)= \mathfrakSp₂^T(S). Moreover, we show that \mathfrakSp₂(S) and \mathfrakSp₂^T(g, n) are completely determined in terms of (g, n).

Citations

• On keen Heegaard splittings
• The classification of Kleinian surface groups, II: The Ending Lamination Conjecture
• The classification of Kleinian surface groups, I: Models and bounds
• Geometry of the complex of curves II: Hierarchical structure

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