# Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $$G$$ of $$\text{Homeo}(S^1)$$, we have the Euler class $$\chi$$ in the group cohomology $$H^2(G;\mathbb{Z})$$. One can think of this class as the pullback of the generator of $$H^2(\mathrm{B}\text{Homeo}(S^1);\mathbb{Z})$$. I have two related questions regarding the (non)vanishing of the powers of Euler class for different subgroups of $$\text{Homeo}(S^1)$$.

1. Does there exist a torsion-free, finitely generated group $$G\subset\operatorname{Homeo}(S^1)$$ such that $$\chi^k$$ for all $$k$$ are nonzero in $$H^{2k}(G;\mathbb{Z})$$?

If we let the group $$G$$ be the group of piecewise linear homeomorphisms of the circle $$\mathbb{R}/\mathbb{Z}$$ which send $$\mathbb{Q}_2/\mathbb{Z}$$ onto itself and have singular points only on $$\mathbb{Q}_2/\mathbb{Z}$$, where $$\mathbb{Q}_2$$ is the ring of dyadic numbers. Ghys and Sergiescu proved, in this case, $$\chi^k$$ for all $$k$$ are nonzero in $$H^{2k}(G;\mathbb{Z})$$ but $$G$$, which is one of the variants of Thompson groups, has many torsion elements.

Another example related to this question is due to Solomon Jekel. Let $$\Gamma_{g}^1$$ be the mapping class group of a surface of genus $$g$$ with a marked point. It is a subgroup of $$\text{Homeo}(S^1)$$. Jekel showed that in this case $$\chi^{g-1}$$ is nonzero in $$H^{2g-2}(\Gamma_{g}^1;\mathbb{Q})$$. Now, one could let $$G$$ be a finite index torsion-free subgroup of $$\Gamma_{g}^1$$. But the power $$g-1$$ is a threshold for the non-vanishing of powers of the Euler class.

1. In the direction of the second example, let $$\text{Tor}_g^1$$ be the Torelli group of the surface of genus $$g$$ and a marked point. What is the threshold $$k$$ for which $$\chi^k$$ is nonzero in $$H^{2k}(\text{Tor}_g^1;\mathbb{Q})$$? Is it less than $$g-1$$?

However at present even the non-triviality of $$e^2 \in H^4(Tor_g^1)$$ seems to be unknown.
• Kupers and I recently (accidentally) answered this: $e^2$ is nontrivial, even on $Tor^1_{g, 1}$. This is Corollary 8.3 of "On the cohomology of Torelli groups". – Oscar Randal-Williams Apr 30 at 17:20