# Dehn filling of hyperbolic 3-manifolds and Gromov volume

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.

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Regarding (1) and (2), yes, it's very possible. Moreover, provided $N$ is hyperbolic, I believe Thurston proves the Gromov norm of $N_1$ is strictly smaller than $N$'s volume. The only time Dehn filling does not strictly lower the Gromov norm is when you're doing a filling on a non-Hyperbolic component (in the JSJ-decomposition) and the incompressible boundary tori of the hyperbolic components remain incompressible after the filling.

I give an example of of this in my "JSJ decompositions of knot and link complements in S^3" paper towards the end. Here it is:

In the picture you see a 3-component link with the components labelled. This link is hyperbolic, but if you delete component 3 (deletion of components is a Dehn filling operation) you get a 2-component link with five incompressible tori and two Whitehead links in the JSJ-decomposition (the other manifolds are Seifert fibred, they are two Trefoils and two manifolds I call "keychain links" which are trivial punctured disc bundles over the circle).

This example shows how you can make arbitrarily extreme examples of the kind you're seeking. You take a very complicated satellite knot or link with many incompressible tori, then you add another component to the link which punctures all the incompressible tori of the original knot/link.

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@Ryan, thank you for your excellent response. It is funny that one weeks ago, I'm carefully reading your paper but I didn't notice this example. (At that time, I am only care about Graph links ) – Bin Yu Jun 17 '13 at 3:32

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.

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@Agol, thank you. Actually you give me a more clear response to the question (2), but sorry that here I only can accept one as answer. – Bin Yu Jun 17 '13 at 3:28