Bounded cohomology of subgroups of mapping class groups

Abstract

We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb-Kaimanovich-Masur rigidity theorem that states that MCG(S) does not contain a higher rank lattice as a subgroup.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper4.abs.html

Cited by in corpus (112)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Asymptotic geometry of the mapping class group and Teichmueller space
• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Induced quasi-cocycles on groups with hyperbolically embedded subgroups
• The mapping class group from the viewpoint of measure equivalence theory
• Groups acting on manifolds: around the Zimmer program
• Geometry of pseudocharacters
• Commensurating endomorphisms of acylindrically hyperbolic groups and applications
• Hyperbolic extensions of free groups
• Big mapping class groups: an overview
• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire
• Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group
• Hyperbolic structures on groups
• Measure equivalence rigidity of the mapping class group
• Constructing group actions on quasi-trees and applications to mapping class groups
• Non simplicité du groupe de Cremona sur tout corps
• Rank-one isometries of proper CAT(0)-spaces
• Signatures of foliated surface bundles and the symplectomorphism groups of surfaces
• Statistics and compression of scl
• Ideal bicombings for hyperbolic groups and applications
• Homomorphisms between mapping class groups
• The topology of $out(F_n)$
• Partial periodic quotient of groups acting on a hyperbolic space
• A hyperbolic Out(F_n)-complex
• Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings
• Geometric inflexibility and 3-manifolds that fiber over the circle
• Cancelation norm and the geometry of biinvariant word metrics
• Relative free splitting and free factor complexes I: Hyperbolicity
• Groups acting acylindrically on hyperbolic spaces
• On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc
• Bounded cohomology with coefficients in uniformly convex Banach spaces
• Small cancellation in acylindrically hyperbolic groups
• Extending higher dimensional quasi-cocycles
• Subgroup decomposition in Out(F_n): Introduction and Research Announcement
• Small cancellation theory over Burnside groups
• Vastness properties of automorphism groups of RAAGs
• Growth of quasiconvex subgroups
• Invariant Orders on Hermitian Lie Groups
• Big mapping class groups with hyperbolic actions: classification and applications
• Translation surfaces and the curve graph in genus two
• Bounded cohomology via partial differential equations, I
• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
• On cofinite subgroups of mapping class groups
• Groups acting on CAT(0) cube complexes with uniform exponential growth
• On stable commutator length in hyperelliptic mapping class groups
• Divergence and quasimorphisms of right-angled Artin groups
• Bounded Cohomology and Deformation Rigidity in Complex Hyperbolic Geometry
• Loxodromic elements in big mapping class groups via the Hooper-Thurston-Veech construction
• A characterization of higher rank symmetric spaces via bounded cohomology
• Right-angled Artin groups as normal subgroups of mapping class groups
• Second bounded cohomology and WWPD
• Random walks, WPD actions, and the Cremona group
• On the structure and arithmeticity of lattice envelopes
• Acylindrical hyperbolicity of groups acting on trees
• Stable commutator length on mapping class groups
• Actions of certain arithmetic groups on Gromov hyperbolic spaces
• The $L^p$-diameter of the group of area-preserving diffeomorphisms of $S^2$
• Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups
• A dynamical characterization of acylindrically hyperbolic groups
• Generic Phenomena in Groups -- Some Answers and Many Questions
• Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms
• Failure of separation by quasi-homomorphisms in mapping class groups
• Maximal Representations of Surface Groups: Symplectic Anosov Structures
• Behavior of geodesic-length functions on Teichmueller space
• Acylindrical hyperbolicity of the three-dimensional tame automorphism group
• Braided Thompson groups with and without quasimorphisms
• Isometry groups of proper CAT(0)-spaces
• On a question of Etnyre and Van Horn-Morris
• Asymptotic shifting numbers in triangulated categories
• Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles
• Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups
• Morse elements in Garside groups are strongly contracting
• On conjugacy separability of fibre products
• Stable commutator length in word-hyperbolic groups
• Bounded cochains on 3-manifolds
• Divergence in lattices in semisimple Lie groups and graphs of groups
• Aut-invariant quasimorphisms on groups
• Bounded cohomology and isometry groups of hyperbolic spaces
• Property $(TT)$ modulo $T$ and homomorphism superrigidity into mapping class groups
• Relative Bounded Cohomology for Groupoids
• Large scale geometry of commutator subgroups
• Bounded Cohomology Classes of Exact Forms
• The Klein bottle has stably unbounded homeomorphism group
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Parabolic subgroups acting on the additional length graph
• Acylindrical hyperbolicity of cubical small-cancellation groups
• Not all finitely generated groups have universal acylindrical actions
• The rates of growth in an acylindrically hyperbolic group
• Infinite-type loxodromic isometries of the relative arc graph
• Markov chains on hyperbolic-like groups and quasi-isometries
• On quasihomomorphisms with noncommutative targets
• Random walks and contracting elements I: Deviation inequality and Limit laws
• Rigidity of mapping class groups mod powers of twists
• On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras
• Counting pseudo-Anosovs as weakly contracting isometries
• Quasi-homomorphisms on mapping class groups vanishing on a handlebody group
• Quasi-isometry type of the metric space derived from the kernel of the Calabi homomorphism
• Bushy pseudocharacters and group actions on quasi-trees
• Harmonic Maps and Teichmueller Theory
• Garside groups and geometry
• Extensions of quasi-morphisms to the symplectomorphism group of the disk
• Signature of surface bundles and bounded cohomology
• Problems on handlebody groups
• Quasi-actions on trees and Property (QFA)
• Median quasimorphisms on CAT(0) cube complexes and their cup products
• The space of non-extendable quasimorphisms
• Isometry groups of proper hyperbolic spaces
• Higher rank hyperbolicity
• 5-dimensional Bourgeois contact structures are tight
• Calabi quasimorphisms for the symplectic ball
• On the entropy norm on $Ham(S^2)$