Non-realizability of the Torelli group as area-preserving homeomorphisms

Nielsen realization problem for the mapping class group Mod(𝑆𝑔) asks whether the natural projection 𝑝𝑔:Homeo+(𝑆𝑔)→Mod(𝑆𝑔) has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion‐free subgroups of Mod(𝑆𝑔) . The main result of this paper is that the Torelli group has no realization inside the area‐preserving homeomorphisms.