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

TL;DR

This paper proves that the Torelli group cannot be realized as a subgroup of area-preserving homeomorphisms, addressing a longstanding question in the Nielsen realization problem for the mapping class group.

Contribution

It demonstrates the non-realizability of the Torelli group within area-preserving homeomorphisms, advancing understanding of the Nielsen realization problem for torsion-free subgroups.

Findings

Torelli group cannot be realized as area-preserving homeomorphisms

Addresses realization problem for torsion-free subgroups of the mapping class group

Provides new insights into the structure of the Torelli group

Abstract

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ 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 $\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms.