Essential Spanning Forests and Electrical Networks on Groups

Abstract

Let Γ be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of Γ, have no cycles, and no finite connected components are called essential spanning forests. The set $$Y$$ of all such subgraphs can be endowed with a compact topology, and G acts on $$Y$$ by translations. We define a “uniform” G-invariant probability measure μ on $$Y$$ show that μ is mixing, and give a sufficient condition for directional tail triviality. For non-cocompact Fuchsian groups we show how μ can be computed on cylinder sets. We obtain as a corollary, that the tail σ-algebra is trivial, and that the rate of convergence to mixing is exponential. The transfer-current function ψ (an analogue to the Green function), is computed explicitly for the Modular and Hecke groups.