As an immediate consequence of Proposition 6.5.2 of Thurston's notes, we have that, if $M$ is a compact 3-manifold with toric boundary and $\tilde M$ is obtained from $M$ via Dehn filling, then for the 'variant' $\|\cdot\|\_0$ of the Gromov norm it holds that $\|[\tilde M,\partial\tilde M]\|\_0\leq \|[M,\partial M]\|\_0$. Under which assumptions is it also true that $\|[\tilde M,\partial\tilde M]\|\leq \|[M,\partial M]\|$, $\|\cdot\|$, being the 'proper' Gromov norm? This holds, for instance, if $M$ is hyperbolic (Lemma 6.5.4), but what if we drop this hypothesis?



Does Dehn filling always decrease Gromov norm?
As mentioned in another answer, one may deduce the desired result from the following facts:


1. (Gromov's Equivalence Theorem): If every component of the boundary of $M$ has amenable fundamental group, then for every $\varepsilon>0$ there exists a fundamental cycle $z$ for $M$ such that $\|z\|\leq \|M,\partial \|+\varepsilon$ and $\|\partial z\|\leq \varepsilon$.
2. (Matsumoto-Morita): If $T$ is a boundary component of $M$ and the fundamental group of $T$ is amenable, then any two fundamental cycles $z,z'$ for $T$ differ by a boundary $c$ such that $\|c\|\leq K\cdot (\|z\|+\|z'\|)$.


Putting together (1) and (2) one easily gets that the simplicial volume is subadditive with respect to gluings along boundaty components having amenable fundamental groups.


Gromov's proof of the Equivalence Theorem is based on the quite technical theory of multicomplexes, which is studied and developed in the preprint by Kuessner cited above. A different proof (using Ivanov's and Monod's approach to bounded cohomology) is given in our recent preprint


<http://arxiv.org/abs/1305.2612>


(joint work with Bucher, Burger, Iozzi, Pagliantini, Pozzetti).


