# 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