## The simplicial volume of closed manifolds covered by $$\mathbb H^2\times \mathbb H^2$$.(English)Zbl 1156.53018

The main purpose of the paper under review is the computation of the Gromov norm of the Riemannian volume form on the product $$\mathbb H^2\times \mathbb H^2$$, where $$\mathbb H^2$$ denotes the hyperbolic plane of constant curvature $$-1$$.
The following main theorem is shown: Let $$\omega_{\mathbb H^2 \times \mathbb H^2} \in H_c^4(PSL_2\mathbb R \times PSL_2\mathbb R)$$ be the image, under the Van Est isomorphism, of the Riemannian volume form on $$\mathbb H^2 \times \mathbb H^2$$. Then $$| | \omega_{\mathbb H^2 \times \mathbb H^2} | | _{\infty}=(2/3) \pi^2$$. Denote by $$| | M| |$$ the simplicial volume introduced by M. Gromov in [Publ. Math., Inst. Hautes Étud. Sci. 56, 5–99 (1982; Zbl 0516.53046)]. Using the main theorem it is shown:
If $$M$$ is a closed, oriented Riemannian manifold whose universal cover $$\tilde M$$ is isometric to $$\mathbb H^2 \times \mathbb H^2$$, then $$\| M\|=3/2\pi^2.Vol(M)$$. Among some corollaries derived from the above result, we would like to mention: Let $$M$$ and $$N$$ be closed, oriented surfaces. Then $$\| M\times N\| =(3/2)\| M\| \| N\|$$. As pointed out by the author this gives the first exact value of a nonvanishing simplicial volume for a manifold not admitting a constant curvature metric. A relevant tool used is the Van Est isomorphism $$\mathbb J :A^*(X,E)^G \to H_c^*(G, E)$$ between the $$G-$$invariant $$E-$$valued differential forms on $$X$$ and the continuous cohomology of $$G$$ with coefficients in $$E$$ where $$E$$ is a $$G$$-module.

### MSC:

 53C20 Global Riemannian geometry, including pinching 55N10 Singular homology and cohomology theory 20J06 Cohomology of groups

Zbl 0516.53046
Full Text:

### References:

 [1] A. Borel N. Wallach Continuous cohomology, discrete subgroups, and representations of reductive groups 2000 2nd edn Providence, RI American Mathematical Society Mathematical Surveys and Monographs · Zbl 0980.22015 [2] Bowen, The Gromov norm of the product of two surfaces, Topology 44 pp 321– (2005) · Zbl 1064.57014 [3] K. Brown Cohomology of groups 1982 New York Springer Graduate Texts in Mathematics [4] M. Bucher T. Gelander Milnor-Wood inequalities for locally ($$\mathbb{H}$$ 2 ) n -manifolds to appear in C. R. Acad. Sci. 2008 [5] Bucher-Karlsson, The proportionality constant for the simplicial volume of locally symmetric spaces, Colloq. Math. 111 ((2)) pp 183– (2008) · Zbl 1187.53042 [6] M. Bucher-Karlsson On minimal triangulations of products of convex polygons Preprint, 2007, to appear in Discrete Comput. Geom [7] Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 pp 233– (1976) · Zbl 0331.55012 [8] Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math 56 pp 5– (1982) [9] A. Guichardet Cohomologie des groupes topologiques et des algèbres de Lie 1980 Paris CEDIC Textes Mathématiques · Zbl 0464.22001 [10] Hirzebruch, Automorphe Formen und der Satz von Riemann-Roch, Symp. Intern. Top. Alg. 1956 pp 129– [11] Ivanov, A canonical cocycle for the Euler class of a flat vector bundle, Dokl. Akad. Nauk 26 pp 78– (1982) · Zbl 0518.57012 [12] S. Kobayashi K. Nomizu Foundations of differential geometry 1963 New York, London Interscience Publishers [13] C. Löh R. Sauer Simplicial volume of Hilbert modular varieties Preprint, arXiv:math/0706.3904v2 [14] N. Monod Continuous bounded cohomology of locally compact groups 2001 Berlin Springer Lecture Notes in Mathematics · Zbl 0967.22006 [15] W. Thurston Geometry and topology of 3-manifolds Lecture Notes (Princeton, Princeton, NJ, 1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.