Finitely presented condensed groups (Q6554690)

The space \(\mathcal{G}\) of finitely generated marked groups is the set of all pairs \((G,A)\), where \(G\) is a group and \(A\) is a finite ordered generating set of \(G\), considered up to a natural equivalence relation. A topology induced by the local convergence of Cayley graphs turns \(G\) into a \(\sigma\)-compact Polish space. Let \(\mathrm{Sub}(G)\) be the set of subgroups of \(G\).\N\NA first result in the paper under review is (Theorem 1.2): For any finitely generated group \(G\), there is an injective, continuous map \(f: \mathrm{Sub}(G) \rightarrow \mathcal{G}\) satisfying the following conditions. (a) \(f\) maps conjugate subgroups of \(G\) to isomorphic marked groups. If \(G\) is finitely presented and \(H \leq G\) is finitely generated, then \(f(H)\) is finitely presented.\N\NThis result allows the author to transfer various topological phenomena occurring in \(\mathrm{Sub}(G)\) to \(\mathcal{G}\). In particular, he provides the first example of a finitely presented group whose isomorphism class in \(\mathcal{G}\) has no isolated points.