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

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 \(k\).


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
Full Text: DOI arXiv


[1] A Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994) 207 · Zbl 0809.53052
[2] R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer (1992) · Zbl 0768.51018
[3] M Bucher-Karlsson, Simplicial volume of locally symmetric spaces covered by \(\mathrm{SL}_3\mathbb R/\mathrm{SO}(3)\), Geom. Dedicata 125 (2007) 203 · Zbl 1124.53025
[4] M Bucher-Karlsson, The simplicial volume of closed manifolds covered by \(\mathbb H^2\times\mathbb H^2\), J. Topol. 1 (2008) 584 · Zbl 1156.53018
[5] S Francaviglia, Hyperbolic volume of representations of fundamental groups of cusped \(3\)-manifolds, Int. Math. Res. Not. (2004) 425 · Zbl 1088.57015
[6] R Frigerio, Commensurability of hyperbolic manifolds with geodesic boundary, Geom. Dedicata 118 (2006) 105 · Zbl 1096.32014
[7] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) · Zbl 0516.53046
[8] M Gromov, I Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. (1988) 93 · Zbl 0649.22007
[9] U Haagerup, H J Munkholm, Simplices of maximal volume in hyperbolic \(n\)-space, Acta Math. 147 (1981) 1 · Zbl 0493.51016
[10] S K Hansen, Measure homology, Math. Scand. 83 (1998) 205 · Zbl 0932.55003
[11] D L Johnson, A short proof of the uniqueness of Haar measure, Proc. Amer. Math. Soc. 55 (1976) 250 · Zbl 0324.43001
[12] D Jungreis, Chains that realize the Gromov invariant of hyperbolic manifolds, Ergodic Theory Dynam. Systems 17 (1997) 643 · Zbl 0882.57009
[13] S Kojima, Polyhedral decomposition of hyperbolic manifolds with boundary, Proc. Workshop Pure Math. 10 (1990) 37
[14] S Kojima, Geometry of hyperbolic \(3\)-manifolds with boundary, Kodai Math. J. 17 (1994) 530 · Zbl 0859.57012
[15] T Kuessner, Efficient fundamental cycles of cusped hyperbolic manifolds, Pacific J. Math. 211 (2003) 283 · Zbl 1066.57026
[16] T Kuessner, Proportionality principle for cusped manifolds, Arch. Math. \((\)Brno\()\) 43 (2007) 485 · Zbl 1199.57012
[17] J F Lafont, B Schmidt, Simplicial volume of closed locally symmetric spaces of non-compact type, Acta Math. 197 (2006) 129 · Zbl 1111.57020
[18] J M Lee, Introduction to smooth manifolds, Graduate Texts in Math. 218, Springer (2003)
[19] C Löh, Measure homology and singular homology are isometrically isomorphic, Math. Z. 253 (2006) 197 · Zbl 1093.55004
[20] C Löh, R Sauer, Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume, J. Topol. 2 (2009) 193 · Zbl 1187.53043
[21] N Peyerimhoff, Simplices of maximal volume or minimal total edge length in hyperbolic space, J. London Math. Soc. \((2)\) 66 (2002) 753 · Zbl 1048.52007
[22] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Math. 149, Springer (1994) · Zbl 0809.51001
[23] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)
[24] A Weil, L’intégration dans les groupes topologiques et ses applications, Actual. Sci. Ind. 869, Hermann et Cie. (1940) 158 · Zbl 0063.08195
[25] A Zastrow, On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory, Pacific J. Math. 186 (1998) 369 · Zbl 0933.55008
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.