The mapping class group from the viewpoint of measure equivalence theory

Abstract

We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover, we give various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, we investigate amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary. Using this investigation, we prove that the mapping class group of a compact orientable surface is exact.

vi+190 pages, 11 figures

References in corpus (3)

• Train tracks and the Gromov boundary of the complex of curves
• Low-dimensional homology groups of mapping class groups: a survey
• Bounded cohomology and isometry groups of hyperbolic spaces

Cited by in corpus (25)

• A survey of Measured Group Theory
• Constructing group actions on quasi-trees and applications to mapping class groups
• OE and W* superrigidity results for actions by surface braid groups
• Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings
• Stationary C*-dynamical systems
• Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups
• Examples of amalgamated free products and coupling rigidity
• Embedding mapping class groups into finite products of trees
• The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces
• Polynomial-time algorithms for the curve graph
• A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• On the top-dimensional ${\ell }^2$-Betti numbers
• Integrable tautness of isometries of complex hyperbolic spaces
• K-theory and actions on Euclidean retracts
• The Novikov Conjecture
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Integrable measure equivalence and rigidity of hyperbolic lattices
• Combinatorics of tight geodesics and stable lengths
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
• Strong primeness for equivalence relations arising from Zariski dense subgroups
• Measure equivalence rigidity of the handlebody groups
• Weak tight geodesics in the curve complex: examples and gaps
• Vanishing of $L^2$-Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings
• Deciding reducibility of mapping classes is in $\text{NP}$