Finitely presented condensed groups

Abstract

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