arXiv subjects

Dean Wardell

Publications and source records attributed to Dean Wardell.

3 recordsLinked to original sources

arXiv2026

Local-global conjugacy questions for affine extensions

Boston and Jones constructed a probabilistic model, called the Markov model, in order to predict the cycle structures of elements in Galois groups $G_n(f)$ associated to the $n$-th iterate of a quadratic postcritically finite polynomial $f$ over a number field. Goksel refined this model, introducing the 'even' Markov groups $M_n(f)$. These groups conjecturally contain a copy of $G_n(f)$, leading to questions about local-global conjugacies within the larger automorphism group $\operatorname{Aut}(T_n)$ of the first $n$ levels of the binary rooted tree coming from arboreal representations of the Galois groups. While the conjugacy results found by Goksel were restricted to the study of these automorphism groups, we generalise the findings to affine extensions of groups by permutation representations, using a cohomological argument. Furthermore, we provide a counterexample to the main conjecture proposed by Goksel, demonstrating that more work is required to resolve the underlying questions regarding Markov models.

math.DS↗

arXiv2025

Profinite geometric iterated monodromy groups of postcritically finite polynomials in degree 3

In this article, we study the properties of profinite geometric iterated monodromy groups associated to polynomials. Such groups can be seen as generic representations of absolute Galois groups of number fields into the automorphism group of a regular rooted tree. Our main result is that, for a degree 3 postcritically finite polynomial over a number field, where each finite postcritical point has at least one preimage outside the critical orbits, the associated profinite geometric iterated monodromy group is finitely invariably generated. Moreover, this group is determined by the isomorphism class of the ramification portrait of the polynomial, up to conjugation by an automorphism of the ternary rooted tree. We also study the group-theoretical properties of such groups, namely their branch and torsion properties. In particular, we show that such groups are regular branch over the closure of their commutator subgroup, and that they contain torsion elements of any order realizable in the ternary tree.

math.DS↗

arXiv2024

An elementary proof of the Benjamini-Nekrashevych-Pete conjecture for the semi-direct products $\mathbb{Z}^n\rtimes \mathbb{Z}$

A finitely generated group $G$ is called strongly scale-invariant if there exists an injective homomorphism $f:G\to G$ such that $f(G)$ is a finite index subgroup of $G$ and such that $\cap_{n\geq 0} f^n(G)$ is finite. Nekrashevych and Pete conjectured that all strongly scale-invariant groups are virtually nilpotent, after disproving a stronger conjecture by Benjamini. This conjecture is known to be true in some situations. Der\'e proved it for virtually polycyclic groups. In this paper, we provide an elementary proof for those polycyclic groups that can be written as a semi-direct product $\mathbb{Z}^n\rtimes \mathbb{Z}$.

math.GR↗