TITLE: Dehn filling  of hyperbolic 3-manifolds and Gromov volume
QUESTION [6 upvotes]: Let $N$ be a hyperbolic 3-manifold with finite (Gromov) volume and with finitely many tori cusps boundary. If we do Dehn fillings along some cusps of $N$, we obtain a new manifold, denoted by $N_1$. Suppose $N_1$ is irreducible, my question is:
(1) is it possible that $N_1$ isn't hyperbolic but contains a hyperbolic JSJ piece?
(2) if (1) is possible, then whether the Gromov norm of $N_1$ is strictly less than the Gromov norm of $N$? 

Remark: By Thurton's book (1978) Proposition 6.5.2, the Gromov norm of $N_1$ is no larger than the Gromov norm of $N$.
Thank you.

REPLY [7 votes]: In Theorem 6.5.6 in Thurston's notes, he indicates that volume decreases strictly under hyperbolic Dehn filling, referring to Theorem 6.4 (volume rigidity, see the appendix to Dunfield's paper for more details http://arxiv.org/abs/math/9802022). I think his proof could be extended to the case of non-hyperbolic Dehn fillings, combining the arguments of 6.4 and 6.5.5, but this hasn't been written down anywhere. 
Alternatively, in my thesis I gave an argument that volume decreases strictly. First, perform an orbifold Dehn filling along the slope to get a hyperbolic Dehn filling, in which volume (and therefore Thurston norm) strictly decreases. Then map this orbifold filling to the Dehn filling along the slope, so that the Gromov norm strictly decreases.