  We investigate Friedl-L\"uck's universal $L^2$-torsion for descending HNN
extensions of finitely generated free groups, and so in particular for
$F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first
cohomology of the group which is an analogue of the Thurston norm for
$3$-manifold groups. We prove that this Thurston semi-norm is an upper bound
for the Alexander semi-norm defined by McMullen, as well as for the higher
Alexander semi-norms defined by Harvey. The same inequalities are known to hold
for $3$-manifold groups. We also prove that the Newton polytopes of the
universal $L^2$-torsion of a descending HNN extension of $F_2$ locally
determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit
means of computing the BNS invariant for such groups. As a corollary, we prove
that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$
has finitely many connected components. When the HNN extension is taken over
$F_n$ along a polynomially growing automorphism with unipotent image in $GL(n,
\mathbb{Z})$, we show that the Newton polytope of the universal $L^2$-torsion
and the BNS invariant completely determine one another. We also show that in
this case the Alexander norm, its higher incarnations, and the Thurston norm
all coincide.
