The simplicial volume of hyperbolic manifolds with geodesic boundary. (English) Zbl 1206.53045

Frigerio, Roberto; Pagliantini, Cristina

Algebr. Geom. Topol. 10, No. 2, 979-1001 (2010).

For any topological space with a fundamental class, we can define its simplicial volume. Oriented manifolds or compact oriented manifolds with boundary admit a fundamental class. If \(M\) is a complete hyperbolic manifold of finite volume, then a celebrated theorem of Thurston says that its simplicial volume is proportional to the hyperbolic volume, i.e., the ratio of the hyperbolic volume of \(M\) over the simplicial volume of \(M\) is equal to the volume \(\nu_n\) of the regular ideal geodesic \(n\)-simplex in the real hyperbolic \(n\)-space, where \(n\) is equal to the dimension of \(M\), \(\text{Vol}(M)/||M||=\nu_n\). On the contrary, following T. Kuessner [Pac. J. Math. 211, No. 2, 283–313 (2003; Zbl 1066.57026)] and D. Jungreis [Ergodic Theory Dyn. Syst. 17, No. 3, 643–648 (1997; Zbl 0882.57009)], for a hyperbolic manifold \(M\) with nonempty totally geodesic boundary \(\partial M\), the corresponding ratio of the volumes is strictly less than \(\nu_n\).
The main result of this paper is that, for any \(\eta>0\), there exists a constant \(k=k(\eta, n)\) such that, if the ratio \(\text{Vol}(\partial M)/\text{Vol}(M)\leq k\), then the ratio of the volumes satisfies \(\text{Vol}(M)/\|M\|\geq\nu_n-\eta\).
This paper also constructs examples of manifolds \(M\) which satisfy the condition \(\text{Vol}(\partial M)/\text{Vol}(M)\leq k\) for arbitrarily small \(k\).

Reviewer: Lizhen Ji (Ann Arbor)

MSC:

53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57N16	Geometric structures on manifolds of high or arbitrary dimension
57N65	Algebraic topology of manifolds

Keywords:

straight simplex; Haar measure; volume form

Citations:

Zbl 1066.57026; Zbl 0882.57009

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