arXiv subjects

Julian Wykowski

Publications and source records attributed to Julian Wykowski.

6 recordsLinked to original sources

arXiv2025

The Profinite Rigidity of Torsion-Free Lamplighter Groups

We prove that the torsion-free lamplighter group $\Gamma = \mathbb{Z}^n \wr \mathbb{Z}$ of any rank $n \in \mathbb{N}$ is profinitely rigid in the absolute sense: the finite quotients of $\Gamma$ determine its isomorphism type uniquely among all finitely generated residually finite groups. The proof combines the theory of profinite rigidity for modules over Noetherian domains with an analysis of the algebraic properties of the lower central series of groups with the same profinite completion as $\Gamma$.

math.GR↗

arXiv2025

The Profinite Rigidity of Free Metabelian Groups

We prove that finitely generated free metabelian groups $\Psi_n$ are profinitely rigid in the absolute sense: they are distinguished by their finite quotients among all finitely generated residually finite groups. The proof is based on a previous result of the author governing profinite rigidity for modules over Noetherian domains, as well as a homological characterisation of free metabelian groups due to Groves--Miller.

math.GR↗

arXiv2025

Profinite Rigidity over Noetherian Domains

We initiate the study of profinite rigidity for modules over a Noetherian domain: to what extent are these objects determined by their finite images? We establish foundational statements in analogy to classical results in the category of groups. We describe three profinite invariants of modules over any Noetherian domain $\Lambda$. We show that free modules are profinitely rigid when $\Lambda$ satisfies a homological condition, and characterise the profinite genus of all modules when $\Lambda$ is a Dedekind domain. In the case where $\Lambda$ is a PID, we find that all finitely generated modules are profinitely rigid. As an application, we prove that solvable Baumslag--Solitar groups are profinitely rigid in the absolute sense. These are the first examples of absolute profinite rigidity among non-abelian one-relator groups and among non-LERF groups.

math.GR↗

arXiv2024

Cohomological Separability of Baumslag--Solitar groups and Their Generalisations

A group $\Gamma$ has separable cohomology if the profinite completion map $\iota \colon \Gamma \to \widehat{\Gamma}$ induces an isomorphism on cohomology with finite coefficient modules. In this article, cohomological separability is decided within the class of generalised Baumslag--Solitar groups, i.e. graphs of groups with infinite cyclic fibers. Equivalent conditions are given both explicitly in terms of the defining graph of groups and in terms of the induced topology on vertex groups. Restricted to the class of Baumslag--Solitar groups, we obtain a trichotomy of cohomological separability and cohomological dimension of the profinite completions. In particular, this yields examples of non-residually-finite one-relator groups which have separable cohomology, and examples which do not.

math.GR↗

arXiv2023

Profinite Subgroup Accessibility and Recognition of Amalgamated Factors

We investigate accessible subgroups of a profinite group $G$, i.e. subgroups $H$ appearing as vertex groups in a graph of profinite groups decomposition of $G$ with finite edge groups. We prove that any accessible subgroup $H \leq G$ arises as the kernel of a continuous derivation of $G$ in a free module over its completed group algebra. This allows us to deduce splittings of an abstract group from splittings of its profinite completion. We prove that any finitely generated subgroup $\Delta$ of a finitely generated virtually free group $\Gamma$ whose closure is a factor in a profinite amalgamated product $\widehat{\Gamma} = \overline{\Delta} \amalg_K L$ along a finite $K$ must be a factor in an amalgamated product $\Gamma = \Delta \ast_\chi \Lambda$ along some $\chi \cong K$. This extends previous results of Parzanchevski--Puder, Wilton and Garrido--Jaikin-Zapirain on free factors.

math.GR↗

arXiv2022

An investigation into Lie algebra representations obtained from regular holonomic D-modules

Beilinson--Bernstein localisation relates representations of a Lie algebra $\mathfrak{g}$ to certain $\mathcal{D}$-modules on the flag variety of $\mathfrak{g}$. In [arXiv:2002.01540], examples of $\mathfrak{sl}_2$-representations which correspond to $\mathcal{D}$-modules on $\mathbb{CP}^1$ were computed. In this expository article, we give a topological description of these and extended examples via the Riemann-Hilbert correspondence. We generalise this to a full characterisation of $\mathfrak{sl}_2$-representations which correspond to holonomic $\mathcal{D}$-modules on $\mathbb{CP}^1$ with at most 2 regular singularities. We construct further examples with more singularities and develop a computer program for the computation of this correspondence in more general cases.

math.AG↗