We prove that the cup product of $Delta$-decomposable quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial. As a corollary we obtain that this is also the case for Brooks quasimorphisms (in particular on selfoverlapping words) and Rolli quasimorphisms.
We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far
more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of classifying spaces and for computing the simplicial volume. In particular, we extend our computations to mapping tori of certain connected sums in higher dimensions. Our main result completes the picture for the vanishing of the simplicial volume of fiber bundles in dimension four. Moreover, we deduce that dimension four together with the trivial case of dimension two are the only dimensions where all mapping tori have vanishing simplicial volume. As a group theoretic consequence, we derive an alternative proof of the fact that the fundamental group $G$ of a mapping torus of a 3-manifold $M$ is Gromov hyperbolic if and only if $M$ is virtually a connected sum $# S^2times S^1$ and $G$ does not contain $mathbb{Z}^2$.
We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.
We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Con
jecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of $pi_1$ to have vanishing simplicial volume. This answers a special case of a question due to Luck.
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.
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, na
mely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.
We prove that for any prime $pgeq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $mathcal{L}_p=(mathbb{Z}/pmathbb{Z})wr mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true
and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $mathcal{L}_2$ is equal to the golden ratio $(1+sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.
A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume |M| of M is equal to vol(M)/v_n, where v_n is a constant depending only on the dimension of M. The same result also holds for
complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio vol(M)/|M| is strictly smaller than v_n if M is compact with non-empty geodesic boundary. We prove here a quantitative version of Jungreis result for n>3, which bounds from below the ratio |M|/vol(M) in terms of the ratio between the volume of the boundary of M and the volume of M. As a consequence, we show that a sequence {M_i} of compact hyperbolic n-manifolds with geodesic boundary is such that the limit of vol(M_i)/|M_i| equals v_n if and only if the volume of the boundary of M_i grows sublinearly with respect to the volume of the boundary of M_i. We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension three.
If $Gamma<mathrm{PSL}(2,mathbb{C})$ is a lattice, we define an invariant of a representation $Gammarightarrow mathrm{PSL}(n,mathbb{C})$ using the Borel class $beta(n)in mathrm{H}^3_mathrm{c}(mathrm{PSL}(n,mathbb{C}),mathbb{R})$. We show that the inva
riant is bounded and its maximal value is attained by conjugation of the composition of the lattice embedding with the irreducible complex representation $mathrm{PSL}(2,mathbb{C})rightarrow mathrm{PSL}(n,mathbb{C})$. Major ingredients of independent interest are the extension to degenerate configuration of flags of a Goncharov cocycle and its study, as well as the identification of $mathrm{H}^3_mathrm{c}(mathrm{SL}(n,mathbb{C}),mathbb{R})$ as a normed space.
We define a notion of semi-conjugacy between orientation-preserving actions of a group on the circle, which for fixed point free actions coincides with a classical definition of Ghys. We then show that two circle actions are semi-conjugate if and onl
y if they have the same bounded Euler class. This settles some existing confusion present in the literature.
