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I have read for example in the introduction of http://arxiv.org/pdf/math/0506338v2.pdf about the property that if we glue hyperbolic manifolds with geodesic boundary consisting of tori along some tori, the volume of the manifold we obtain is simply the sum of the volumes of the pieces.

The only reference i found for this result is Thurston's book Geometry and topology of 3-manifolds.

However, Proposition 6.5.2 of the book seems to prove only an inequality. I think the other inequality might be a consequence of Theorem 6.5.5 in the same book but the proof of this result is missing.

Therefore i would like to find a precise reference for this property of additivity of the simplicial volume.

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  • $\begingroup$ A small error in what you write -- a torus cannot appear in the geodesic boundary of a hyperbolic manifold. Perhaps you mean cusp. $\endgroup$
    – Sam Nead
    Commented Nov 24, 2014 at 17:32
  • $\begingroup$ Thilo Kuessner's thesis should have a self-contained account. However, it does not appear to be easily available on-line. Perhaps contact him directly. $\endgroup$
    – Sam Nead
    Commented Nov 24, 2014 at 17:44
  • $\begingroup$ Thilo Kuessner's thesis "Relative simplicial volume" can be found here publikationen.uni-tuebingen.de/xmlui/handle/10900/48322 $\endgroup$
    – j.c.
    Commented Feb 28, 2018 at 20:17

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You can't have geodesic boundary tori on a hyperbolic manifold: the tori arise as cusp sections. However, it is true in general that if you glue two 3-manifolds along incompressible tori then the simplicial volume of the resulting manifold is just the sum of the simplicial volumes of the pieces.

More generally, simplicial volume is additive whenever you glue two manifolds along boundary components that are $\pi_1$-injective and have amenable funamental group. You can find a proof there.

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  • $\begingroup$ Sorry for the mistake about tori boundary, the correct statement i should have written is the one you make. Thank very much for this reference! $\endgroup$
    – Renaud Detcherry
    Commented Nov 27, 2014 at 12:27

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