The structure of group-labeled graphs forbidding an immersion

A $\mathrm{\Gamma }$-labeled graph is an oriented graph with edges invertibly labeled by a group $\mathrm{\Gamma }$. We prove a structure theorem for $\mathrm{\Gamma }$-labeled graphs which forbid a fixed $\mathrm{\Gamma }$-labeled graph as an immersion, for any finite $\mathrm{\Gamma }$. Roughly, we show that such graphs admit a tree-cut decomposition in which every bag either contains few high degree vertices or is nearly signed over a proper subgroup of $\mathrm{\Gamma }$.