Finitely presented condensed groups

Abstract

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to isomorphic marked groups; in addition, if $G$ is finitely presented and $H\le G$ is finitely generated, then $f(H)$ is finitely presented. This result allows us to transfer various topological phenomena occurring in $Sub(G)$ to $\mathcal G$. In particular, we provide the first example of a finitely presented group whose isomorphism class in $\mathcal G$ has no isolated points.