4 problems

Let $F_2$ be the free group of rank two, and let $\operatorname{CO}(F_2)$ denote its space of circular orders. The group-action space…

Let $G$ be a countable group. Write $\operatorname{CO}(G)$ for the space of circular orders on $G$, and let $\operatorname{Hom}(G,\operatorname{Homeo}_+(S^1))$ be the space of acti…

Free-product no-isolated-points conjecture. The space $\operatorname{CO}(G*H)$ either has no isolated points or is finite. In particular, if it is infinite, then…

Kleinian-group characterization conjecture. There should be a similar characterization of Kleinian groups by faithful, orientation-preserving topological actions on $S^1$ with thre…