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
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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)
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