Mapping class group orbit closures for non-orientable surfaces

Abstract

Let S be a connected non-orientable surface with negative Euler characteristic and of finite type. We describe the possible closures in ML and PML of the mapping class group orbits of measured laminations, projective measured laminations and points in Teichmüller space. In particular we obtain a characterization of the closure in ML of the set of weighted two-sided curves.

The limit set of non-orientable mapping class groups

“…A few days after the first version of this paper was posted on arXiv, the author was notified of another recent paper by Erlandsson, Gendulphe, Pasquinelli, and Souto [3] which proves the Conjecture 9.1 of [6], i.e., a stronger version of Theorem 3.3. The techniques they use are significantly different, relying on careful analysis of train track charts carrying various measured laminations.…”

“…In particular, for certain foliations, the dynamics of the first return map do not fully capture the dynamics of the foliation. Therefore, the train track chart analysis in [3] is stronger.…”

“…While this paper was being written, neither the author, nor the authors of [3] were aware of each others' work.…”

The limit set of non-orientable mapping class groups

“…Another paper on mapping class group orbit closures. A few days after the first version of this paper was posted on arXiv, the author was notified of another recent paper by Erlandsson, Gendulphe, Pasquinelli, and Souto [Erl+21] which proves the Conjecture 9.1 of [Gen17], i.e. a stronger version of Theorem 3.3.…”

“…The techniques they use are significantly different, relying on careful analysis of train track charts carrying various measured laminations. While this paper was being written, neither the author, nor the authors of [Erl+21] were aware of each others' work.…”