The Gromov invariant of negatively curved manifolds

Simplicial volume of ℚ–rank one locally symmetric spaces covered by the product of ℝ–rank one symmetric spaces

“…Since X i has negative sectional curvature bounded away from zero for all i D 1; : : : ; k , the volume of geodesic n i -simplices in X i is uniformly bounded from above for all i D 1; : : : ; k (see Inoue and Yano [13]). Hence, there exists a uniform bound on the volume of geodesic n-simplices in X i for all i D 1; : : : ; k .…”

“…First, the positivity of the simplicial volume was verified for closed negatively curved manifolds by Thurston [22], Gromov [12] and Inoue and Yano [13]. It was verified for closed locally symmetric spaces covered by SL 3 .R/=SO 3 .R/ by Savage [21] and Bucher-Karlsson [3].…”

Simplical volume of closed locally symmetric spaces of non-compact type

“…Gromov cites the work of Thurston [18], Inoue-Yano [13] and Savage [16], in which positivity is verified for compact manifolds that are real hyperbolic, negatively curved and locally symmetric quotients of SL(n, R)/SO(n, R), respectively. Moreover, Gromov [10] shows that characteristic classes may be represented using bounded cohomology classes implying positivity for several additional classes of locally symmetric spaces.…”

Simplicial volume of locally symmetric spaces covered by $${\rm SL}_{3}{\mathbb{R}}/{\rm SO}(3)$$

“…In [6], Inoue and Yano generalize ideas of Thurston [10] in order to show that the volume of geodesic simplices in any (fixed) symmetric space of real rank one is uniformly bounded, thus proving Theorem 2 in this case. In [7] the simplices in consideration are constructed with the barycenter method, and the obtained volume bound strongly relies on previous work by Connell and Farb [3].…”