Bounded cohomology and non-uniform perfection of mapping class groups

Representation stability for the cohomology of the pure string motion groups

The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups given by Jensen, McCammond, and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational co…

Vanishing of cohomology groups of random simplicial complexes

We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of the…

Bounded Cohomology of Finitely Generated Kleinian Groups

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…

Barycentric straightening and bounded cohomology

We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G, in degrees n-r+2 and higher, every cohomology class has a bounded representative. This answers Dupont's problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension n-r have unbounded volume, showing that t…

Thin buildings

Let X be a building of uniform thickness q+1. L^2-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L^2-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis complex of the Coxeter group is a manifold, a version of Poincare duality allows to deduce that the L^2-cohomology of a building with large thickness is concentrated in the top dimension.

Analysis and Geometry of Boundary-Manifolds of Bounded Geometry

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the boundary) then its first relative L^2-cohomology vanishes (this in particular answers a question of Roe). We prove the Hodge-de Rham-theorem for L^2-cohomology of oriented boundary-manifolds of bounded g…

Norms on the cohomology of hyperbolic 3-manifolds

We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, Şengün, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-…

单纯有界上同调与稳定性

Simplicial bounded cohomology and stability

本文引入组合技术研究半单纯集、单纯复形和偏序集的单纯有界上同调，证明新的有界无环性结果，并推导出两大类线性群有界上同调的斜率-1/2稳定性结果。

伊斯马吉洛夫定理的$C^0$类似物

$C^0$-Analogue of Ismagilov's Theorem

该研究建立了保体积微分同胚的伊斯马吉洛夫定理的$C^0$类似物，证明了保体积同胚群单位分支的一阶连续上同调与一阶德拉姆上同调同构，解决了保体积$C^0$通量猜想。

带与不带拟同态的辫化汤普森群

Braided Thompson groups with and without quasimorphisms

本研究分析了多种辫化汤普森群的拟同态与有界上同调，证明$bV$等群有无穷维拟同态空间，而$\bV$二阶有界上同调平凡，提供了相关特殊群类的首个实例。