Simplicial volume, barycenter method, and bounded cohomology

Homological norms on nonpositively curved manifolds

“…The idea of simplex straightening was first developed by Gromov [21] and Thurston [30] in negative curvatures as a tool for obtaining lower bounds on the Gromov norm (also see [22]). This method has been extended by many authors in different contexts including higher rank symmetric spaces [23,25,26,31], certain nonpositively curved manifolds [12,13] and others [28]. Below we give a brief overview on some of these results.…”

Higher genera for proper actions of Lie groups

“…In fact, the conjecture in [9] is that for semisimple Lie groups all these cocycles are bounded. For recent work on this conjecture, see [13,21]. In this last reference, different simplices are used, given by the barycentric subdivision of the geodesic ones, to prove boundedness of the top dimensional cocycle for general connected semisimple Lie groups.…”

“…We remark that item (2) above has in independent interest, and should be compared with the literature on bounded cohomology of Lie groups, c.f. [13,21] The geometric applications stated in Theorem 1.5 are then a direct consequence of the G homotopy invariance of the signature index class, established by Fukumoto in [10] and, for the higher A-genera, of the vanishing of the index class Ind C * r (G) (ð) ∈ K * (C * r (G)) of the spin Dirac operator ð of a G-spin G-proper manifold endowed with a G-metric of positive scalar curvature, established by Guo, Mathai and Wang in [11]. In the odd dimensional case we argue by suspension.…”

On splitting rank of non-compact-type symmetric spaces and bounded cohomology

“…One of the key step in their approach is to show boundedness of a certain Jacobian, which relies heavily on previous work of Connell and Farb [6], [7]. Recently, Inkang Kim and Sungwoon Kim [17] extended the Jacobian estimate to codimension one (but the codimension one surjectivity of the comparison map is automatic), and they also gave detailed investigation on rank two cases. Meanwhile, Lafont and Wang [20] showed surjectivity in codimesion ≤ rank(X) − 2, in irreducible cases excluding SL(3, R) and SL (4, R).…”