  In the 1970s, Fathi, having proven that the group of compactly supported
volume-preserving homeomorphisms of the $n$-ball is simple for $n \ge 3$, asked
if the same statement holds in dimension $2$. We show that the group of
compactly supported area-preserving homeomorphisms of the two-disc is not
simple. This settles what is known as the "simplicity conjecture" in the
affirmative. In fact, we prove the a priori stronger statement that this group
is not perfect.
  An important step in our proof involves verifying for certain smooth twist
maps a conjecture of Hutchings concerning recovering the Calabi invariant from
the asymptotics of spectral invariants defined using periodic Floer homology.
Another key step, which builds on recent advances in continuous symplectic
topology, involves proving that these spectral invariants extend continuously
to area-preserving homeomorphisms of the disc. These two properties of PFH
spectral invariants are potentially of independent interest.
  Our general strategy is partially inspired by suggestions of Fathi and the
approach of Oh towards the simplicity question. In particular, we show that
infinite twist maps, studied by Oh, are not finite energy homeomorphisms, which
resolves the "infinite twist conjecture" in the affirmative; these twist maps
are now the first examples of Hamiltonian homeomorphisms which can be said to
have infinite energy. Another consequence of our work is that various forms of
fragmentation for volume preserving homeomorphisms which hold for higher
dimensional balls fail in dimension two.
