The Geometry of Profinite Graphs with Applications to Free Groups and Finite Monoids

We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition. We define a pseudovariety of groups H to be arboreous if all finitely generated free pro-H groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties H, a pro-H analog of the Ribes and Zalesskĭı product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions H to the much studied pseudovariety equation J ∗H = J m © H.