Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $Γ$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a higher rank symmetric space iff $H^2_b(M;\R)\to H^2(M;\R)$ is injective (and otherwise the kernel is infinite-dimensional). This is the converse of a theorem of Burger-Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements.
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
Bounded cohomology of groups was first studied by Gromov in 1982. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic `non-positively curved' groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension 2 and 3 in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by Fujiwara and Kapovich.
We show that in the bounded cohomology of non-abelian free groups the Massey triple product is always trivial when the second factor is represented by the coboundary of a decomposable quasi-morphism. We also show that in the bounded cohomology of a negatively curved compact Riemannian manifold the Massey triple product is always trivial when the second factor is represented by an exact differential form.
We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and construct a finitely presented non-amenable boundedly acyclic group. On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to~$2$, and a concrete finitely presented one. Countable non-amenable groups with these two extre…
It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL_2 over S-integers.
Let $M$ be a complete connected Riemannian manifold of finite volume. In this paper we present a new method of constructing classes in bounded cohomology of transformation groups such as $Homeo_0(M,μ)$, $Diff_0(M,vol)$ and $Symp_0(M,ω)$ (in case $M$ is symplectic). As an application we show that, under certain conditions on $π_1(M)$, the $3^{rd}$ bounded cohomology of these groups is infinite dimensional.
We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed …
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic gro…
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.
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.
It was proved by Mineyev and Yaman that, if $(Γ, Γ')$ is a relatively hyperbolic pair, the comparison map $$ H_b^k(Γ, Γ'; V) \to H^k(Γ, Γ'; V) $$ is surjective for every $k \ge 2$, and any bounded $Γ$--module $V$. By exploiting results of Groves and Manning, we give another proof of this result. Moreover, we prove the opposite implication under weaker hypotheses than the ones required by Mineyev and Yaman.
We generalize the result of Brandenbursky and Marcinkowski for the bounded cohomology of transformation groups to infinite volume case. To state the result, we introduce the notion of norm controlled cohomology as a generalization of bounded cohomology. This cohomology theory also provides a framework for relative quasimorphisms.
We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.
Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic $3$-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a $2$ dimensional subspace of bounded cohomology. Our techniques apply to…
We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we have a group with infinitely many ends has infinite dimensional second bounded cohomology.
Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded cohomology started to develop in 1982, thanks to the pioneering paper "Volume and Bounded Cohomology" by M. Gromov, where the definition of bounded cohomology was extended to deal also with topological spaces. The aim of this monograph is to provide an introduction to bounded cohomology of discrete groups and of topological spaces. We also describe some applications of the theory to related active resear…
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that, contrary to ordinary cohomology, the diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism gro…
We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of bounded rank.
We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded cohomology of a pair of topological spaces is isometrically isomorphic to the bounded cohomology of the pair of fundamental groupoids. Furthermore, we prove that bounded cohomology relative to an amenable groupoid is isometrically isomorphic to the bounded cohomology of the ambient groupoid.
We give a new perspective on the homological characterisations of amenability given by Johnson in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group.
We prove that the cup product of $Δ$-decomposable quasimorphisms, Brooks quasimorphisms or Rolli quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial.
This paper is devoted to the computation of the space $H_b^2(Γ,H;\mathbb{R})$, where $Γ$ is a free group of finite rank $n\geq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $Γ$ if and only if $H_b^2(Γ,H;\mathbb{R})$ is not trivial, and furthermore, if and only if there is an isometric embedding $\oplus_\infty^n\mathcal{D}(\mathbb{Z})\hookrightarrow H_b^2(Γ,H;\mathbb{R})$, where $\mathcal{D}(\mathbb{Z})$ is the space of bounded alternating functions on $\mathbb{Z}$ equipped with the defect norm.
We define a transfer map in the setting of bounded cohomology with certain metric G-module coefficients. As an application, we extend a theorem of Chatterji, Mislin, Pittet and Saloff-Coste on the comparison map from Borel-bounded to Borel cohomology, to cover the case of Lie groups with finitely many connected components.
We prove that there is a bound on the dimension of the first cohomology group of a finite group with coefficients in an absolutely irreducible in characteristic p in terms of the sectional p-rank of the group.
We investigate bounds on the dimension of cohomology groups for finite groups acting on an irreducible kG-module for G a finite group of bound sectional p-rank and k an algebraically closed field of characteristic p.
We establish the vanishing for non-trivial unitary representations of the bounded cohomology of SL_d up to the rank. It holds more generally for uniformly bounded representations on superreflexive spaces. The same results are obtained for lattices. We also prove that the real bounded cohomology of any lattice is invariant in the same range.
In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of $\mathrm{IA}_3$, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the low…
We study the $\mathrm{Out}(F_n)$-action on the second bounded cohomology $H^2_b(F_n, \mathbb{R})$, focusing on the countable-dimensional dense invariant subspace given by Brooks quasimorphisms. We show that this subspace has no finite-dimensional invariant subspaces, in particular no fixpoints, partially answering a question of Miklós Abért. To this end we introduce a notion of speed of an element $g\in \mathrm{Out}(F_n)$, which measures the asymptotic growth rate of bounded cohomology classes under repeated application of $g$.
We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2, R) in degree four, establishing a special case of a conjecture of Monod.
We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.
For a group $G$, we construct a quasi morphism from its left orderings and the map from the space of left orderings to the second bounded cohomology. We show that these maps reflect various properties of the group orderings.
We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.
We prove surjectivity of the comparison map from continuous bounded cohomology to continuous cohomology for Hermitian Lie groups with finite center. For general semisimple Lie groups with finite center, the same argument shows that the image of the comparison map contains all the even generators. Our proof uses a Hirzebruch type proportionality principle in combination with Gromov's results on boundedness of primary characteristic classes and classical results of Cartan and Borel on the cohomology of compact homogeneous spaces.
We study the bounded cohomology of certain groups acting on the Cantor set. More specifically, we consider the full group of homeomorphisms of the Cantor set as well as Thompson's group $V$. We prove that both of these groups are boundedly acyclic, that is the bounded cohomology with trivial real coefficients vanishes in positive degrees. Combining this result with the already established $\mathbb{Z}$-acyclicity of Thompson's group $V$, will make $V$ the first example of a finitely generated group, in fact the first example of a group of type $F_\infty$, which is universally boundedly acyclic.…
A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of $\mathbb{R}^n$ (Matsumoto--Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as comp…
Dehn quandles of surfaces and their bounded cohomology
Indian Institute of Science Education and Research (IISER) Mohali（印度科学教育与研究学院（IISER）莫哈利分校）；Indian Institute of Science Education and Research (IISER) Tirupati（印度科学教育与研究学院（IISER）蒂鲁帕蒂分校）
The theory of bounded cohomology of groups has many applications. A key open problem is to compute the full bounded cohomology $H_b^n(F, R)$ of a non-abelian free group $F$ with trivial real coefficients. It is known that $H_b^n(F,R)$ is trivial for $n=1$ and uncountable dimensional for $n=2,3$, but remains unknown for any $n \geq 4$. For $n=4$, one may construct classes by taking the cup product $α\cup β\in H_b^4(F, R)$ between two $2$-classes $α, β\in H^2_b(F, R)$. However, we show that all such cup products are trivial if $α$ and $β$ are classes induced by the quasimorphisms defined by Broo…
We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type II_1 equivalence relations with trivial outer automorphism group that are mutually…
On open covers of aspherical spaces satisfying $π_1$-constraints and bounded Adamson cohomology
Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg（马丁·路德·哈雷-维滕贝格大学数学研究所）
