Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces [Asymptotique sous-dominante des invariants spectraux d’entrelacs et groupes d’hom´eomorphismes de surfaces]

In previous work, we defined “link spectral invariants” for any compact surface and used these to study the algebraic structure of the group of area-preserving homeomorphisms; in particular, we showed that the kernel of Fathi’s mass-flow homomorphism is never simple. A key idea for this was a kind of Weyl law, showing that asymptotically the link spectral invariants recover the classical Calabi invariant.

In the present work, we use the subleading asymptotics in this Weyl law to learn more about the algebraic structure of these homeomorphism groups in the genus zero case. In particular, when the surface has boundary, we show that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple, answering an old question of Oh and M¨uller; this contrasts the smooth case, where the kernel of Calabi is simple. We similarly show that the group of hameomorphisms of the two-sphere is not simple. Related considerations allow us to extend the Calabi homomorphism to the full group of compactly supported area-preserving homeomorphisms, answering a longstanding question of Fathi. In fact, we produce infinitely many distinct extensions. Central to the applications is that we show that the subleading asymptotics for smooth, possibly time-dependent, Hamiltonians are always Op1q, and for certain autonomous maps recover the Ruelle invariant. The construction of a hameomorphism with “infinite Ruelle invariant” then shows that a normal subgroup with Op1q subleading asymptotics is proper.