The generating graph of a profinite group

Abstract

Let $G$ be 2-generated group. The generating graph $\Gamma(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a 2-generated profinite group $G,$ considering in this case topological generation. We prove that the set $V(G)$ of non-isolated vertices of $\Gamma(G)$ is closed in $G$ and that, if $G$ is prosoluble, then the graph $\Delta(G)$ obtained from $\Gamma(G)$ by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinite group $G$ with the property that $\Delta(G)$ has $2^{\aleph_0}$ connected components. This implies that the so called "swap conjecture" does not hold for finitely generated profinite groups. We also prove that if an element of $V(G)$ has finite degree in the graph $\Gamma(G),$ then $G$ is finite.