Essential spanning forests and electric networks in groups

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 ${\cal Y}$ of all such subgraphs being given a compact topology, G acts on ${\cal Y}$ continuously. We define a G-invariant measure $\mu$ on ${\cal Y}$ and investigate ergodic properties of the process $({\cal Y},\ \mu,\ G),$ called an ESF.The case G = ${\rm {\bf Z}}\sp{d}$ or their finite extensions was studied by Pemantle and Burton. For a general G we establish mixing and give a sufficient condition for directional tail triviality in terms of the transfer-current function $\psi .$ For non-co-compact Fuchsian groups we establish the tail triviality of $({\cal Y},\ \mu,\ G)$ and describe $\psi,$ which by a theorem of Burton and Pemantle determines the measure $\mu.$The second part of this thesis is concerned with the relation between ESF and algebraic dynamical systems. Burton and Pemanle computed the entropy of $({\cal Y},\ {\bf Z}\sp{d}),$ where ${\cal Y}$ is the set of essential spanning forests in an arbitrary ${\bf Z}\sp{d}$-periodic graph. Their formula turned out to be the same as the one obtained by Lind, Schmidt and Ward for the entropy of ${\bf Z}\sp{d}$-actions by automorphisms on certain compact subgroups of $({\bf R/Z})\sp{{\bf Z}\sp{d}}.$ We give a direct proof of equalities of entropies for ESF processes and corresponding algebraic systems, answering the question of the above authors.