The virtually generating graph of a profinite group

TL;DR

This paper explores the structure and connectivity of graphs derived from finitely generated profinite groups, revealing how these graphs' properties relate to the groups' generating sets and subgroup structures.

Contribution

It introduces and analyzes new graphs associated with profinite groups, demonstrating their connectivity properties and the existence of groups with prescribed graph components.

Findings

The graph $ ilde riangle_{ m{virt}}(G)$ is connected with diameter at most 3.

For any positive integer t, there exists a finitely generated prosoluble group with exactly t connected components in $ riangle_{ m{virt}}(G)$.

The graph $ riangle_{ m{virt}}(G)$'s connectivity depends on the topological generation of open subgroups.

Abstract

We consider the graph $\Gamma_{\rm{virt}}(G)$ whose vertices are the elements of a finitely generated profinite group $G$ and where two vertices $x$ and $y$ are adjacent if and only if they topologically generate an open subgroup of $G$. We investigate the connectivity of the graph $\Delta_{\rm{virt}}(G)$ obtained from $\Gamma_{\rm{virt}}(G)$ by removing its isolated vertices. In particular we prove that for every positive integer $t$, there exists a finitely generated prosoluble group $G$ with the property that $\Delta_{\rm{virt}}(G)$ has precisely $t$ connected components. Moreover we study the graph $\tilde \Gamma_{\rm{virt}}(G)$, whose vertices are again the elements of $G$ and where two vertices are adjacent if and only if there exists a minimal generating set of $G$ containing them. In this case we prove that the subgraph $\tilde \Delta_{\rm{virt}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.