I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is topologizable iff there exists a topology on $G$ which makes it into a Hausdorff non-discrete topological group.

Main question:Is every infinite amenable group topologizable?

Main motivation for this question is that perhaps naively non-topologizability seemed to me such a strange property that I hoped it could be used to show existence of non-sofic groups.

Question:Is there an infinite non-topologizable sofic group?

After failing to prove that sofic groups are topologizable I thought it would be still interesting to prove that infinite "elementary sofic" groups are topologizable. Elementary sofic groups are for the purpose of this discussion the class of "groups which are provably sofic by current methods", i.e. it contains all amenable groups, is closed under taking free products amalgamated over amenable groups, extensions with sofic kernel and amenable quotient, /any other results which are in the literature/, and with the property that if G is residually elementary sofic then G is elementary sofic.

Question:Is there an infinite non-topologizable elementary sofic group?

Unfortunately my plan to answer the above question negatively failed at step 1, and hence the Main question.