Towards Ivanov's meta-conjecture for geodesic currents

TL;DR

This paper explores automorphism groups of geodesic currents and measured laminations on surfaces, confirming Ivanov's meta-conjecture in most cases and highlighting challenges in the broader context.

Contribution

It proves that $Aut(ML)$ is isomorphic to the extended mapping class group, supporting Ivanov's meta-conjecture, and constructs examples illustrating difficulties for $Aut(\mathscr{C})$.

Findings

$Aut(ML)$ is isomorphic to the extended mapping class group in most cases.

Constructs infinite families of curves with identical length spectra and intersection numbers.

Highlights the complexity of proving Ivanov's conjecture for $Aut(\mathscr{C})$.

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.