Ergodicity conjecture for non-extreme Euler-class components

About 21 years old · traced to

Let Σ\Sigma be a closed surface of genus gg, let G=PSL(2,R)G=\mathsf{PSL}(2,\mathbb{R}), and let ee be the Euler-class map on the representation space Hom(π,G)\mathsf{Hom}(\pi,G). For each integer kk with 1≤k≤2g+b−21\leq k\leq 2g+b-2, consider the component e−1(2−2g+b+k)e^{-1}(2-2g+b+k).

Euler-component ergodicity conjecture. For each integer 1≤k≤2g+b−21\leq k\leq 2g+b-2, the ModΣ\mathsf{Mod}_{\Sigma}-action on the component e−1(2−2g+b+k)e^{-1}(2-2g+b+k) of Hom(π,G)\mathsf{Hom}(\pi,G) is ergodic.

The claim concerns the non-extreme components of the PSL(2,R)\mathsf{PSL}(2,\mathbb{R}) representation space, contrasting with the proper action on the extreme components corresponding to Teichmüller space. The source does not state whether this conjecture has been resolved.

References

Primary source

William M. Goldman, “Mapping Class Group Dynamics on Surface Group Representations”, arXiv:math/0509114 (2006).

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