Recent zbMATH articles in MSC 22B05https://zbmath.org/atom/cc/22B052024-03-13T18:33:02.981707ZWerkzeugSettled elements in profinite groupshttps://zbmath.org/1528.370802024-03-13T18:33:02.981707Z"Cortez, María Isabel"https://zbmath.org/authors/?q=ai:cortez.maria-isabel"Lukina, Olga"https://zbmath.org/authors/?q=ai:lukina.olgaAn infinite rooted tree \(T\) is called a \(d\)-ary tree if it contains a single vertex called the root at level \(0\), and for all \(n\geq 1\), each vertex at level \(n-1\) is joined to \(d\geq 2\) vertices at level \(n\). Denote by \(\mathrm{Aut}(T)\) the set of automorphisms on \(T\). Let \(\sigma\in \mathrm{Aut}(T)\). Then \(\sigma\) restricted on the set of vertices at level \(n\) is a permutation of a set of \(d^n\) elements. A vertex \(v\) at level \(n\) is said to be in a cycle of length \(k \geq 1\) of \(\sigma\) if \(\sigma^k(v) = v\) and \(\sigma^j(v)\neq v\) for every \(1\leq j <k\). It is in a stable cycle if all the descendants of \(\{v, \sigma(v), \dots, \sigma^{k-1}(v)\}\) at level \(m>n\) form a cycle of length \(d^{m-n}k\). We say that \(\sigma\) is settled if the number of vertices in stable cycles at level \(n\) divided by \(d^n\) tends to \(1\). A profinite subgroup \(\mathcal{G}\subset\mathrm{Aut}(T)\) is called densely settled if the set of settled elements in \(\mathcal{G}\) is dense in \(\mathcal{G}\).
Let \(K\) be an algebraic number field (containing \(\mathbb{Q}\)), and \(\overline{K}\) a separable closure of \(K\). For \(\alpha\in K\) and a polynomial \(f\) of degree \(d\) with coefficients in \(K\), we can associate a \(d\)-ary tree \(T\) by choosing \(V_n=\{f^{-n}(\alpha)\}\) as vertices at level \(n\) and taking an edge joining a vertex \(\beta\in V_{n+1}\) and \(\gamma\in V_n\) if \(f(\beta)=\gamma\). An arboreal representation of \(K\) is a homomorphism \(\rho_{f,\alpha}: \mathrm{Gal}(\overline{K}/K) \rightarrow\mathrm{Aut}(T)\), from the Galois group \(\mathrm{Gal}(\overline{K}/K)\) to the space \(\mathrm{Aut}(T)\) of automorphisms on the \(d\)-ary tree \(T\). Let \(Y_n\) be the Galois group of the extension \(K(f^{-n}(\alpha))\). Then the image of the representation is the profinite groupe \(Y_{\infty}=\varprojlim \{Y_{n+1} \rightarrow Y_n\} \).
In a series of papers [Geom. Dedicata 124, 27--35 (2007; Zbl 1206.11069); Pure Appl. Math. Q. 5, No. 1, 213--225 (2009; Zbl 1167.11011); Proc. Am. Math. Soc. 140, No. 6, 1849--1863 (2012; Zbl 1243.11115)], \textit{N. Boston} and \textit{R. Jones} studied the density of settled elements in the image of an arboreal representation and conjectured that for a quadratic polynomial \(f\) and \(\alpha\in \mathbb{Q}\), the set of settled elements is dense in the image of \(\rho_{f,\alpha}\).
The present paper studies the conjecture of N. Boston and R. Jones for quadratic polynomials having a strictly pre-periodic post-critical orbit of length \(2\). Among others, the authors prove that the above defined profinite groupe \(Y_{\infty}\) is densely settled. The authors also provide new evidence that the conjecture of Boston and Jones holds for quadratic polynomials with strictly pre-periodic post-critical orbits of length at least \(3\).
The main tools of the paper are maximal abelian subgroups and their Weyl groups. The novelty is an application of the Weyl group framework to the study of arboreal representations and groups acting on trees.
Reviewer: Lingmin Liao (Créteil)