Let $G$ be a non-abelian group and $\mathcal S$ and $\mathcal T$ be group topologies on $G$. What is the largest group topology $\tau$ on $G$ with $\tau \subseteq \mathcal T\cap \mathcal S$?

In abelian case it is easy to find a base of neighborhoods around $1$ for $\tau$. In this paper there are some propositions abut infimum of two field topologies. But I could not find a general investgation about infimum of two group topologies.