Let $\|M\|$ denote the simplicial volume of an oriented closed connected manifold.
If $f : M \to N$ is a continuous map between two oriented closed connected manifolds of the same dimension, then $|\deg f|\|N\| \leq \|M\|$. In particular, if $M$ admits a self-map $f$ with $|\deg f| > 1$, then $\|M\| = 0$. Is the converse true? I suspect that the answer is no, but I haven't been able to find a counterexample.
Is there an oriented closed connected manifold $M$ with $\|M\| = 0$ but which does not admit a self-map $f$ with $|\deg f| > 1$?
For instance, if $M$ is a compact Nilmanifold, it has zero simplicial volume. On the other hand, there are nontrivial finitely-generated torsion-free co-hopfian nilpotent groups $\Gamma$ (see for instance here). Taking $M$ a compact Nilmanifold with the fundamental group $\Gamma$ we obtain an example of a manifold with zero simplicial volume an no self-maps of degree $>1$. (Any such map would induce an epimorphism $\Gamma\to \Gamma'$, where $\Gamma'<\Gamma$ is a finite index subgroup. That would contradict the co-hopfian property of $\Gamma$.