Simplicial volume of Hilbert modular varieties

Details Simplicial volume of Hilbert modular varieties Simplicial volume of Hilbert modular varieties

2007-06-26

arxiv.org/abs/0706.3904v3

Mathematics

Geometric Topology

11 pages; rearrangement of section and minor changes; final version; to appear in Commentarii Mathematici Helvetici

Scientific paper

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact locally symmetric spaces of finite volume, to so-called Hilbert modular varieties. The key idea is to reduce the problem to the compact case by first relating the simplicial volume of these manifolds to the Lipschitz simplicial volume and then taking advantage of a proportionality principle for the Lipschitz simplicial volume. Moreover, using computations of Bucher-Karlsson for the simplicial volume of products of closed surfaces, we obtain the exact value of the simplicial volume of Hilbert modular surfaces.

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