Automorphism groups of graphs as topological groups

Percolation on Transitive Graphs as a Coalescent Process: Relentless Merging Followed by Simultaneous Uniqueness

“…Let |A| denote the cardinality of a set A. For equivalent definitions of unimodularity, see Trofimov [31] and Benjamini, Lyons, Peres and Schramm [7]. Most quasi-transitive graphs that come up naturally are unimodular.…”

“…In particular, the Cayley graph of any finitely generated group is transitive and unimodular. A transitive graph adding an edge between x and its ξ-grandparent; see [31] or [7]. In this example, p u = 1, and for p ∈ (p c , 1) every infinite cluster C has a unique vertex v(C, ξ) that is "closest" to ξ.…”

Approximating graphs with polynomial growth

“…Gromov's structure theorem has been extended to vertex transitive graphs with polynomial growth by [10]. This can also be seen as a special case of Losert's [8] classi®cation of topological groups with polynomial growth (cf.…”

Indistinguishability of Percolation Clusters

“…By Benjamini, Lyons, Peres and Schramm (1999) [hereinafter referred to as BLPS (1999)], a transitive graph is unimodular iff there is some unimodular transitive closed subgroup of Aut(G). It is not hard to show that a transitive closed subgroup r c Aut(G) is unimodular iff for all x, y E V(G), we have I{z E V(G): 3 y E r yx = x and yy = z} I = I {z E V ( G): 3 y E r y y = y and y x = z} I [see Trofimov (1985)]. …”