Bounded cohomology of subgroups of mapping class groups (Hyperbolic Spaces and Discrete Groups)

By M. Bestvina, K. Fujiwara. 2000

115 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/34ec134ef5a8872648398342dee2e9f6159c9a13

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quasi-convexity of hyperbolically embedded subgroups (2013, result): “rthere exists Rso that for any x;y2Hwith d(x;y) Rthere are only nitely many g2Gso that d(x;gx);d(y;gy) r. It has to be mentioned that the weak proper discontinuity property of Bestvina and Fujiwara [BF02] is very similar to the acylindricity condition we gave above. First of all, let us show that th”
Quasi-morphisms on surface diffeomorphism groups (2019, methodology): “If gis a hyperbolic element then any orbit of gis a C-quasi-axis, i.e. a g-invariant C-quasi-geodesic, for some Cdepending on the orbit. To establish our main theorem we use the following notion from [BF02]. Denition 2.4 ([BF02]). Let g 1;g 2 2Gbe two hyperbolic elements with a C-quasi-axis A 1 and ”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, background): “, mapping class groups enjoy several properties reminiscent of nonpositively/negatively curved spaces, including: automaticity (and, thus, quadratic Dehn function) [Mos95], having many quasimorphisms [BF02], super-linear divergence [Beh06], etc. Mapping class groups also exhibit coarse versions of s”
Hyperbolic structures on groups (2017, background): “For example, it should work for actions considered in [9].”
Contracting elements and random walks (2011, background): “cting if for some (and hence any) x 0 2Xand any integer n (1)the orbit of x 0 is a quasi-geodesic, (2)all special paths connecting x 0 to gn(x 0) are PS contracting with uniform constant. Recall from [BFu1] that, given a group Gacting on the metric space X, the element g2Gis WPD if the orbits of hgi”
Induced quasicocycles on groups with hyperbolically embedded subgroups (2012, methodology): “perly on a proper CAT(0) space and let gbe a rank-1 element. Then gis contained in the unique maximal virtually cyclic subgroup E(g) of Gand E(g) is hyperbolically embedded in G[26]. Example 1.6 (cf. [6, 8]). Let us illustrate our theorem by extending quasimorphisms in the case when Gand gare as in ”
Tight geodesics in the curve complex (2008, methodology): “Proof: (cf. [ BesF ]) If we realise a, b ∈ Σ so as to minimise the number of intersections, then a ∪ b, as a combinatorial graph embedded in Σ ,i s determined up to isotopy.”
Divergence and quasimorphisms of right-angled Artin groups (2010, background): “Supp(h′) )Supp(c), as claimed. For the second statement of the theorem, note that the action of any non-cyclic subgroup of AΓ on XΓ is weakly properly discontinuous in the sense of Bestvina– Fujiwara [BF1] since it is a properly discontinuous action of a non-virtually cyclic groupon a CAT(0) space. ”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “ssume that G acts of T without inversions, then the condition |V(T )| ≥ 3 in the lemma can be relaxed to |V(T )| ≥ 2. The next result is obtained in the course of proving part (3) of Proposition 6 in [17]. Note that although the authors assume that every hyperbolic element satisﬁes WPD there, the pr”