The proportionality constant for the simplicial volume of locally symmetric spaces

Abstract

Abstract. We follow ideas going back to Gromov's seminal article [Publ. Math. IHES 56 (1982)] to show that the proportionality constant relating the simplicial volume and the volume of a closed, oriented, locally symmetric space M = Γ \G/K of noncompact type is equal to the Gromov norm of the volume form in the continuous cohomology of G. The proportionality constant thus becomes easier to compute. Furthermore, this method also gives a simple proof of the proportionality principle for arbitrary manifolds.

The simplicial volume of closed manifolds covered by ℍ2× ℍ2

“…Observe that by Step 2, both f 2 and δh 3 vanish on 5-tuples z satisfying n 1 (z) + n 2 (z) 7, so that the same holds for f 3 . We will now prove, step by step, that f 3 also vanishes on 5-tuples z with (n 1 (z), n 2 (z)) = (3, 5), (4,4), (4,5), and (5,5). In all but one subcase, the strategy is the same as in most of the proof of Step 2.…”

“…. , (x 4 , y 4 )) is equal to 1 120 σ ∈Sym (5) sign(σ)Or(x σ (0) , x σ (1) , x σ (2) ) · Or(y σ (2) , y σ (3) , y σ (4) ).…”

“…Recall that the simplicial volume M of a closed, oriented manifold M is a topological invariant introduced by Gromov in [8] and is defined as As an immediate consequence of our main theorem, we obtain in Theorem 1 the explicit proportionality constant relating the simplicial volume and the volume of closed, oriented, Riemannian manifolds covered by H 2 × H 2 . Indeed, we prove in [5,Theorem 2] that the proportionality constant of the proportionality principle for a closed, oriented locally symmetric space of noncompact type M n = Γ\G/K is precisely the Gromov norm of the volume form in H n c (G, R). Explicit finite values of the proportionality constant were previously known only for hyperbolic manifolds [8,15]: in which case it is equal to the maximum volume of ideal geodesic simplices in H n , a constant that has been computed explicitly only up to dimension n = 6.…”

Volume invariant and maximal representations of discrete subgroups of Lie groups

“…Bucher-Karlsson [5] reformulates a proof of Gromov's proportionality principle in the language of continuous bounded cohomology and moreover, shows that…”

Stable complexity and simplicial volume of manifolds

“…We denote by v n the volume of the ideal regular hyperbolic simplex in H n . The following result is due to Thurston [46] and Gromov [19] (detailed proofs can be found in [3,7,43] for the closed case, and in [5,12,15,16] for the cusped case).…”