Invariant Radon measures on measured lamination space

Abstract

Let S be a non-exceptional oriented surface of finite type. We classify all Radon measures on the space of measured geodesic laminations for S which are invariant under the mapping class group.Comment: Writing improved. 41 pages, 1 figur

A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

“…For any subsurface R, there is a natural inclusion g * (I c R (α)) that is still locally finite and ergodic, but is now invariant under the whole mapping class group. Mirzakhani and Lindenstrauss prove that every MCG-invariant locally finite ergodic measure on MF is a multiple of µ T h or µ R,c T h for some c and R. The same result was obtained in [Ham09].…”

Spectral rigidity of automorphic orbits in free groups

“…The space Curr.F N / of all geodesic currents on F N comes equipped with a natural weak-topology making it into a locally compact space, and with a natural left Out.F N /-action by linear transformations. The theory of geodesic currents on free groups has been actively developed in the last several years by Kapovich [35;36;37;38] and 40;42;41] (see also Bestvina and Feighn [3], Clay and Pettet [11], Hamenstädt [32], Francaviglia [28] and Kapovich and Nagnibeda [44;45;46] for other recent applications of currents). The space Curr.F N / turns out to be a natural counterpart for the Outer space cv N , and, more generally, the closure cv N of cv N .…”

Geodesic Currents and Counting Problems

“…We prove next that any such limit ν is supported by the space ML of measured laminations (Lemma 4.2), and that it gives measure 0 to the set of non-filling laminations (Lemma 4.4). It then follows from Lindenstrauss-Mirzakhani's classification of locally finite mapping class group invariant measures on ML [12,11] that the limit ν is a multiple c · µ Thu of the Thurston measure (Lemma 4.5). All that remains is to show that c = m(α) m Σ .…”