Towards Ivanov's meta-conjecture for geodesic currents

TL;DR

This paper studies automorphism groups Aut($\mathscr{C}$) and Aut($\mathcal{ML}$) preserving the intersection form on a closed surface $S$, in the spirit of Ivanov's meta-conjecture. It proves Aut($\mathcal{ML}$) is isomorphic to $Mod^{\pm}(S)$ for all genera with the genus-2 case modulo the hyperelliptic involution, and shows Aut($\mathscr{C}$) surjects onto $Mod^{\pm}(S)$ but may have a nontrivial kernel, as demonstrated by constructed pairs of curves with the same self-intersection and simple marked length spectrum. The approach hinges on establishing linearity, preservation of simple closed curves and their weights, and relating Aut($\mathcal{ML}$) to the curve complex automorphisms via Ivanov's theorem, while highlighting obstructions in passing from laminations to currents. The results illuminate when Ivanov's meta-conjecture holds for laminations and outline precise obstacles for currents, contributing to understanding automorphism groups of surface-associated geometric structures and their connection to the mapping class group.

Abstract

Given a closed, orientable surface $S$ of negative Euler characteristic, we study two automorphism groups: $Aut(\mathscr{C})$ and $Aut(\mathcal{ML})$, groups of homeomorphisms that preserve the intersection form in the space $\mathscr{C}$ of geodesic currents and the space $\mathcal{ML}$ of measured laminations. We prove that except in a few special cases, $Aut(\mathcal{ML})$ is isomorphic to the extended mapping class group. This theorem is a special case of \textit{Ivanov's meta-conjecture}. We investigate this question for $Aut(\mathscr{C})$. To demonstrate the difficulty in proving Ivanov's conjecture for $Aut(\mathscr{C})$, we construct infinite family of pairs of closed curves that have the simple same marked length spectra and self intersection number.

Figures (5)

• Figure 1: Two types of complimentary region of $|\lambda|$. Here, $\gamma$ is a peripheral curve of $S'$
• Figure 2: Example of a lamination $k$ that intersects $\gamma$ but not $\lambda$.
• Figure 3: Closed curve with number of half twists $a =19$, $b = 8$ and $c = 5$. The figure shows the curve $\gamma_{(19,8,5)}$ intersecting itself minimally.
• Figure 4: Arcs $s_1$, $s_2$ and $s_3$ are homotoped to intersect $\gamma_{(a,b,c)}$ minimal number of times.
• Figure 5: (Top) Shaded region is the image of the immersion $f: P \rightarrow S'$. The boundary curves $X$, $Y$, $Z$ are mapped to closed curves on $S'$ of the respective color. (Bottom right) $\alpha$ and $\beta$ are simple closed curves on $S'$ that intersect $f(P)$. (Bottom left) The arcs in $P$ are inverse images of $\alpha$ and $\beta$ in $P$.

Theorems & Definitions (40)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• Lemma 3.4
• proof