# The profinite and pro-p genus of free and surface groups.

• Ismael Morales (University of Oxford)
• Friday 10 February 2023, 13:45-14:45
• MR13.

Let $S$ be a free or surface group. A finitely generated group $G$ is said to be in the profinite (or pro-$p$) genus of $S$ if it is residually-finite (resp. residually-$p$) and has the same collection of quotients in the class of finite groups (resp. finite $p$—groups). It is an open question whether the profinite genus of S uniquely consists of the group $S$. Nevertheless, the pro—$p$ genus is bigger than the profinite genus in these cases, and we will see how this can be used to confirm a weaker version of the question. We also handle the similar case of $S\times \mathbb{Z}n$. Extending this result to a bigger class of groups would require improvements of Lück’s approximation principles for the $L2$ invariants and Gromov’s conjecture about the existence of surface subgroups.

This talk is part of the Geometric Group Theory (GGT) Seminar series.