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Essential Spanning Forests and Electrical Networks on Groups
Authors:Rita Solomyak
Institution:(1) School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
Abstract:Let Gamma be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of Gamma, 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 ldquouniformrdquo G-invariant probability measure mgr on 
$$Y$$
show that mgr is mixing, and give a sufficient condition for directional tail triviality. For non-cocompact Fuchsian groups we show how mgr can be computed on cylinder sets. We obtain as a corollary, that the tail sgr-algebra is trivial, and that the rate of convergence to mixing is exponential. The transfer-current function psgr (an analogue to the Green function), is computed explicitly for the Modular and Hecke groups.
Keywords:probability measure  Cayley graph  spanning forest  non-cocompact Fuchsian group
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