The geometry of profinite graphs with applications to free groups and finite monoids

Abstract

Abstract. 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 Zalesskiȋ 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.

On power groups and embedding theorems for relatively free profinite monoids

“…(1 •1) This work was extended in [31]. A characterization of all solutions to (1•1) was obtained by the authors in [6].…”

“…In particular, the only known non-solutions were pseudovarieties of Abelian groups, where inequality is obvious. The main result of this paper is that the solutions to (1•2) are precisely what are called Hall pseudovarieties in [6,31]. These are exactly the pseudovarieties of groups H for which the notions of being H-extendible (in the sense of Margolis, Sapir and Weil [18]) and being H-closed coincide for finitely generated subgroups of a free group.…”

“…For a (profinite) monoid M generated by a (profinite) set A, we use Γ A (M ) to denote its (profinite) Cayley graph; the reader is referred to [3,6,9] for basic notions concerning profinite graphs. Recall that for an A-generated group, the monoid M (G) denotes the Meakin-Margolis [15] expansion of G. It is an A-generated inverse monoid and consists of all pairs (X, g) where g ∈ G and X is a connected finite subgraph of Γ A (G), containing 1 and g. The projection M (G) → G, (X, g) → g is a morphism and the induced congruence on M (G) is usually denoted σ.…”

“…These are exactly the pseudovarieties of groups H for which the notions of being H-extendible (in the sense of Margolis, Sapir and Weil [18]) and being H-closed coincide for finitely generated subgroups of a free group. In [6], it is shown that this is necessary for being a non-trivial solution to (1•1). Thus, all non-trival solutions to (1•1) are solutions to (1•2).…”

“…The set of all solutions to (1•2) shares many of the properties of the set of all non-trivial solutions to (1•1): they form a right ideal in the monoid of pseudovarieties of groups with M [7]; any extension-closed pseudovariety is a solution [28,30]; any pseudovariety defined by a pseudoidentity of the form x π = 1 with π an infinite supernatural number is a solution [6]; they are join-irreducible; and they satisfy no non-trivial group identities. For example, The proof uses a mixture of techniques.…”