Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?

The question is inspired by and related to this one.

Surely, globally non-topologizable infinite groups do exist.

• I don't know, but the class of groups admitting no such topology is closed under taking subgroups, because if $H\subset G$ we can extend the topology on $H$ so that $G\smallsetminus H$ is open and discrete. In particular, a group as in your question will be torsion, and the question boils down to finitely generated groups. – YCor Aug 23 '14 at 14:44

Indeed, Olshanskii's group $G$ has exponent $p^2$ and the cyclic centre of order $p$. In addition, $g^p\ne1$ for any non-central element of the group. So, if we have a topology on $G$ and the multiplication is continuous at $1$, then the function $f\colon x\mapsto x^p$ is also continuous at $1$. But $f$ takes only finitely many values ($p$ values). Thus, this continuous at $1$ function to a discrete (finite) set must be constant on a neighbourhood of $1$ and, hence, this neighbourhood must be finite (because the $f(x)=1$ only for central $x$).